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PROFESSOR: I really stated
everything about discrete
00:00:27.140 --> 00:00:29.560
coding as clearly as I
could in the notes.
00:00:29.560 --> 00:00:33.410
I stated it again as clearly
as I could in class.
00:00:33.410 --> 00:00:36.950
I stated it again as clearly
as I could in making up
00:00:36.950 --> 00:00:40.470
problems that would illustrate
the ideas.
00:00:40.470 --> 00:00:42.820
If I talked about it again
it would just be totally
00:00:42.820 --> 00:00:43.490
repetitive.
00:00:43.490 --> 00:00:49.290
So, at this point, if you want
to understand things better,
00:00:49.290 --> 00:00:52.230
you gotta come up with specific
questions and I will
00:00:52.230 --> 00:00:56.370
be delighted to deal
with them.
00:00:56.370 --> 00:00:58.020
So we want to go on--.
00:00:58.020 --> 00:01:01.580
Oh, there's one other thing
I wanted to talk about.
00:01:01.580 --> 00:01:05.570
We're not having a new problem
set out today.
00:01:05.570 --> 00:01:07.170
I don't think most of you would
00:01:07.170 --> 00:01:09.650
concentrate on it very well.
00:01:14.430 --> 00:01:18.590
I'll tell you what the problems
are, which will be
00:01:18.590 --> 00:01:21.180
due on October 14th.
00:01:24.380 --> 00:01:25.870
I'll pass it out later.
00:01:28.950 --> 00:01:47.120
It's problems 1 through 7, and
the end of lectures 8 to 10,
00:01:47.120 --> 00:01:51.060
and one other one all the way
at the end, problem 26.
00:01:55.670 --> 00:01:59.470
So, 1, 2, 3, 4, 5,
6, 7 and 26.
00:02:05.270 --> 00:02:10.020
So you can get started on them
whenever you choose and I'll
00:02:10.020 --> 00:02:15.590
pass out a traditional problem
set form next time.
00:02:21.680 --> 00:02:25.920
Last time we started to talk
about the difference between
00:02:25.920 --> 00:02:29.930
Reimann and Lebesgue
integration.
00:02:29.930 --> 00:02:33.810
Most people tell me this is
further into mathematics then
00:02:33.810 --> 00:02:35.990
I should go.
00:02:35.990 --> 00:02:41.430
If you agree with them after
we spend a week or two on
00:02:41.430 --> 00:02:44.980
this, please let me know and
I won't torture future
00:02:44.980 --> 00:02:46.940
students with it.
00:02:46.940 --> 00:02:51.920
My sense is that in the things
we're going to be dealing with
00:02:51.920 --> 00:02:55.900
for most of the rest of the
term, knowing a little bit
00:02:55.900 --> 00:03:00.340
extra about mathematics is going
to save you an awful lot
00:03:00.340 --> 00:03:04.100
of time worrying about trivial
little things that you
00:03:04.100 --> 00:03:06.240
shouldn't be worrying about.
00:03:06.240 --> 00:03:09.190
In other words, the great
mathematicians of the 19th
00:03:09.190 --> 00:03:13.780
century who developed --
00:03:13.780 --> 00:03:16.470
yeah, the 19th century, but
partly the 20th century.
00:03:16.470 --> 00:03:20.990
These mathematicians were really
engineers at heart.
00:03:20.990 --> 00:03:25.190
The 20th century and the 21st
century, mathematics has very
00:03:25.190 --> 00:03:32.680
much become very separated from
physics and applications.
00:03:32.680 --> 00:03:36.970
But in the 19th century and
in the early 20th century,
00:03:36.970 --> 00:03:42.630
mathematicians and physicists
were almost the same animals.
00:03:42.630 --> 00:03:45.550
You could scratch one and you'd
find another one there.
00:03:45.550 --> 00:03:50.990
They were very much interested
in dealing with real things.
00:03:50.990 --> 00:03:55.300
They were like the very best of
mathematicians everywhere
00:03:55.300 --> 00:03:58.060
and the very best of engineers
everywhere.
00:03:58.060 --> 00:04:01.830
They really wanted to make life
simpler instead of making
00:04:01.830 --> 00:04:03.960
life more complicated.
00:04:03.960 --> 00:04:06.590
One way that many people
express it is that
00:04:06.590 --> 00:04:11.100
mathematicians are lazy, and
because they're lazy, they
00:04:11.100 --> 00:04:14.270
don't want to go through a lot
of work, and therefore, they
00:04:14.270 --> 00:04:16.770
feel driven to simplify
things.
00:04:16.770 --> 00:04:19.520
There's an awful lot
of truth in that.
00:04:19.520 --> 00:04:22.450
What we're going to learn here
I think will, in fact,
00:04:22.450 --> 00:04:27.580
simplify what you know about
Fourier series and Fourier
00:04:27.580 --> 00:04:29.650
integrals a great deal.
00:04:29.650 --> 00:04:33.480
Engineers typically don't worry
about those things.
00:04:33.480 --> 00:04:36.490
An awful lot of engineers, and
unfortunately, even those who
00:04:36.490 --> 00:04:41.280
write books often state theorems
and leave out the
00:04:41.280 --> 00:04:45.130
last clause of the theorem, and
the last clause of most of
00:04:45.130 --> 00:04:49.110
those theorems the only way to
make them true is to add the
00:04:49.110 --> 00:04:53.960
clause or not, as the case may
be, at the end of it, which
00:04:53.960 --> 00:04:56.330
makes what they state absolutely
meaningless,
00:04:56.330 --> 00:05:00.590
because anything you can add
or not, as the case may be,
00:05:00.590 --> 00:05:02.640
and it becomes true
at that point.
00:05:02.640 --> 00:05:05.550
The whole question becomes well,
what are those cases
00:05:05.550 --> 00:05:07.450
under which it's true.
00:05:07.450 --> 00:05:11.930
That's what we're going to deal
with a little bit here.
00:05:11.930 --> 00:05:16.410
We're going to say just enough
so you start to understand
00:05:16.410 --> 00:05:20.670
these major things that cause
problems, and I hope you will
00:05:20.670 --> 00:05:23.450
get to the point where you don't
have to worry about them
00:05:23.450 --> 00:05:25.590
after that.
00:05:25.590 --> 00:05:27.890
Well, the first thing, we
started talking about it last
00:05:27.890 --> 00:05:31.920
time, we were talking about the
difference between Reimann
00:05:31.920 --> 00:05:34.250
integration and Lebesgue
integration.
00:05:34.250 --> 00:05:37.430
Reimann was a great
mathematician, but he came
00:05:37.430 --> 00:05:41.480
before all of these
mathematicians who were
00:05:41.480 --> 00:05:45.320
following Lebesgue, and who
started to deal with all of
00:05:45.320 --> 00:05:48.870
the problems with this classical
integration, which
00:05:48.870 --> 00:05:54.530
started to fall apart
throughout
00:05:54.530 --> 00:05:56.550
most of the 19th century.
00:05:56.550 --> 00:06:02.130
What Reimann said, well, split
up the axis into equal
00:06:02.130 --> 00:06:06.430
intervals, then approximate
the function within each
00:06:06.430 --> 00:06:13.410
interval, add up all of those
approximate values, and then
00:06:13.410 --> 00:06:18.680
let this quantization over the
time axis become finer and
00:06:18.680 --> 00:06:21.060
finer and finer.
00:06:21.060 --> 00:06:24.630
If you're lucky, you will
come to a limit.
00:06:24.630 --> 00:06:27.060
You can sort of see when you
get to a limit and when you
00:06:27.060 --> 00:06:28.310
don't get to a limit.
00:06:28.310 --> 00:06:31.190
If the function is smooth enough
and you break it up
00:06:31.190 --> 00:06:34.610
finely enough, you're going to
get a very good approximation,
00:06:34.610 --> 00:06:36.970
and if you break it up more
and more finely, the
00:06:36.970 --> 00:06:39.470
approximation gets better
and better.
00:06:39.470 --> 00:06:43.835
If the function is very wild, if
it jumps around wildly, and
00:06:43.835 --> 00:06:48.190
we'll look at examples of that
later, then this doesn't work.
00:06:48.190 --> 00:06:50.690
We'll see why, in fact, this
approach does work.
00:06:50.690 --> 00:06:55.700
Lebesgue said, no, instead of
quantizing along this axis,
00:06:55.700 --> 00:06:59.510
quantize on this axis.
00:06:59.510 --> 00:07:05.190
So he said, OK, start with a
zero, quantize into epsilon, 2
00:07:05.190 --> 00:07:10.430
epsilon, 3 epsilon and so forth,
and everybody when they
00:07:10.430 --> 00:07:12.150
talk about epsilon
they're thinking
00:07:12.150 --> 00:07:14.080
about something small.
00:07:14.080 --> 00:07:16.720
They're also thinking about
making it smaller and smaller
00:07:16.720 --> 00:07:20.270
and smaller and hoping that
something nice happens.
00:07:20.270 --> 00:07:24.780
So then what he said is after
you quantize this axis, start
00:07:24.780 --> 00:07:29.090
to look at how much of the
function lies in each one of
00:07:29.090 --> 00:07:31.410
those little windows.
00:07:31.410 --> 00:07:36.290
I've drawn that out here, mu 2
is the amount of the function
00:07:36.290 --> 00:07:41.420
that lies between 2 epsilon
and 3 epsilon.
00:07:41.420 --> 00:07:45.600
Now the function lies between
2 epsilon and 3 epsilon
00:07:45.600 --> 00:07:49.520
starting at this point, which
I've labeled t1, going up to
00:07:49.520 --> 00:07:53.410
this point, which I've labeled
t2, on over here.
00:07:53.410 --> 00:07:57.260
It's not in this interval
until we get back to t3.
00:07:57.260 --> 00:08:00.460
It stays in this interval
until t4.
00:08:00.460 --> 00:08:04.940
Lebesgue said, OK, let's say
that the function is between 2
00:08:04.940 --> 00:08:11.790
epsilon and 3 epsilon over a
region of size, t2 minus t1,
00:08:11.790 --> 00:08:16.310
which is this size interval,
plus t4 minus t3, which is
00:08:16.310 --> 00:08:18.690
this size interval.
00:08:18.690 --> 00:08:22.800
Instead of saying size, he said
size gets too confusing
00:08:22.800 --> 00:08:26.010
so I'll call it measure
instead.
00:08:26.010 --> 00:08:28.450
That was the beginning of
measure theory, essentially.
00:08:28.450 --> 00:08:30.020
Well, in fact other people
talked about
00:08:30.020 --> 00:08:31.490
measure before Lebesgue.
00:08:31.490 --> 00:08:34.220
There are a lot of famous
mathematicians who were all
00:08:34.220 --> 00:08:37.690
involved in doing this.
00:08:37.690 --> 00:08:41.505
Anyway, that's the basic
idea of what measure
00:08:41.505 --> 00:08:42.690
is concerned with.
00:08:42.690 --> 00:08:47.370
Now, for this curve here, it's
a nice smooth curve, and you
00:08:47.370 --> 00:08:50.160
can almost see intuitively that
you're going to get the
00:08:50.160 --> 00:08:55.490
same thing looking at it this
way or looking at it this way.
00:08:55.490 --> 00:08:58.810
In fact, you do.
00:08:58.810 --> 00:09:02.730
Anyway, what he finally wound
up with is after saying what
00:09:02.730 --> 00:09:06.120
the measure was on each one of
those little slices, how much
00:09:06.120 --> 00:09:08.810
of the function lay in each
one of those intervals, he
00:09:08.810 --> 00:09:10.630
would just add them all up.
00:09:10.630 --> 00:09:13.160
He would add up how much of
the function was in each
00:09:13.160 --> 00:09:16.370
slice, he would multiply how
much of the function was in a
00:09:16.370 --> 00:09:20.480
slice by how high the
slice was up, and
00:09:20.480 --> 00:09:21.980
then he'd get an answer.
00:09:21.980 --> 00:09:26.490
One difference between what he
did and what Reimann did was
00:09:26.490 --> 00:09:29.020
that he always got a lower bound
in doing it this way.
00:09:29.020 --> 00:09:32.310
If he was dealing with a
non-negative function, his
00:09:32.310 --> 00:09:35.360
approximation was always a
little less than what the
00:09:35.360 --> 00:09:39.550
function was, because anything
that lay between 2 epsilon and
00:09:39.550 --> 00:09:43.410
3 epsilon, he would approximate
it as 2 epsilon,
00:09:43.410 --> 00:09:46.180
which is a little less than
numbers between 2
00:09:46.180 --> 00:09:47.760
epsilon and 3 epsilon.
00:09:47.760 --> 00:09:52.550
So this is a lower bound,
whereas this is whatever it
00:09:52.550 --> 00:09:55.830
happens to be -- however you
decide to approximate the
00:09:55.830 --> 00:10:02.960
function there, and there are
lots of ways of doing it.
00:10:02.960 --> 00:10:05.460
Well, I won't prove any of these
things, I just want to
00:10:05.460 --> 00:10:11.010
point them out so that when you
get frustrated with this,
00:10:11.010 --> 00:10:13.730
you can always rely on this.
00:10:13.730 --> 00:10:16.220
Which says that whenever the
Reimann integral exists, in
00:10:16.220 --> 00:10:19.290
other words, whenever the
integral that you're used to
00:10:19.290 --> 00:10:25.260
exists, mainly, whenever it has
meaning, Lebesgue integral
00:10:25.260 --> 00:10:27.850
gives you the same value.
00:10:27.850 --> 00:10:31.220
In other words, you haven't lost
anything by going from
00:10:31.220 --> 00:10:33.910
Reimann integration to
Lebesgue integration.
00:10:33.910 --> 00:10:36.380
You can only gain,
you can't lose.
00:10:36.380 --> 00:10:39.340
The familiar rules for
calculating Reimann integrals
00:10:39.340 --> 00:10:42.330
also apply for Lebesgue
integrals.
00:10:42.330 --> 00:10:44.790
You remember what all those
rules are, you probably know
00:10:44.790 --> 00:10:48.520
all of them better even than
this fundamental definition of
00:10:48.520 --> 00:10:52.810
an integral, which is split up
the function into tiny little
00:10:52.810 --> 00:10:56.940
increments, because throughout
all of the courses that you've
00:10:56.940 --> 00:10:59.830
taken, learning about
integration, learning about
00:10:59.830 --> 00:11:02.650
differentiation, learning about
all of these things,
00:11:02.650 --> 00:11:05.740
what you've done for the most
part is to go through
00:11:05.740 --> 00:11:08.720
exercises using these
various rules.
00:11:08.720 --> 00:11:12.870
So, you know how to integrate
lots of traditional functions.
00:11:12.870 --> 00:11:15.710
You memorized what the integral
of many of them is.
00:11:15.710 --> 00:11:18.830
You had many ways of combining
them to find out what the
00:11:18.830 --> 00:11:21.550
integral of many other
functions is.
00:11:21.550 --> 00:11:25.190
If you program a computer to
calculate these integrals, a
00:11:25.190 --> 00:11:27.600
computer can do it both ways.
00:11:27.600 --> 00:11:29.950
It can either use all the rules
to find out what the
00:11:29.950 --> 00:11:34.000
value of an integral is, or it
can chop things up finely and
00:11:34.000 --> 00:11:36.800
find out that way.
00:11:36.800 --> 00:11:43.120
As I said before, if you think
that being able to calculate
00:11:43.120 --> 00:11:46.870
integrals is what engineering
is about, think again.
00:11:46.870 --> 00:11:48.880
I told you before you
could be replaced
00:11:48.880 --> 00:11:51.270
by a digital computer.
00:11:51.270 --> 00:11:52.780
It's worse than that.
00:11:52.780 --> 00:11:57.200
You could be replaced by
your handheld Palm,
00:11:57.200 --> 00:11:58.790
whatever it's called.
00:11:58.790 --> 00:12:02.720
You can be replaced
by anything.
00:12:02.720 --> 00:12:05.240
After a while we're going to
wear little things embedded in
00:12:05.240 --> 00:12:06.945
our body that will tell
us when we're sick
00:12:06.945 --> 00:12:08.330
and things like this.
00:12:08.330 --> 00:12:12.620
You can be replaced by
those things even.
00:12:12.620 --> 00:12:17.230
So you really want to learn
more than just that.
00:12:17.230 --> 00:12:20.090
For some very weird functions,
the Lebesgue integral exists,
00:12:20.090 --> 00:12:22.470
but the Reimann integral
doesn't exist.
00:12:22.470 --> 00:12:27.070
Why do we want to worry
about weird functions?
00:12:27.070 --> 00:12:28.820
I want to tell you two
reasons for it --
00:12:28.820 --> 00:12:30.870
I told you about it last time.
00:12:30.870 --> 00:12:36.730
One thing is an awful lot of
communication theory is
00:12:36.730 --> 00:12:41.050
concerned with going back and
forth between the time domain
00:12:41.050 --> 00:12:43.820
and the frequency domain.
00:12:43.820 --> 00:12:47.940
When you talk about things which
are straightforward in
00:12:47.940 --> 00:12:52.720
the time domain, often they
become very, very weird in the
00:12:52.720 --> 00:12:54.940
frequency domain
and vice versa.
00:12:54.940 --> 00:13:00.270
I'm going to give you a
beautiful example of that
00:13:00.270 --> 00:13:01.330
probably in the next class.
00:13:01.330 --> 00:13:03.540
You can look at it --
00:13:03.540 --> 00:13:06.660
it is in the appendix.
00:13:06.660 --> 00:13:09.630
You'll see this absolutely weird
function which has a
00:13:09.630 --> 00:13:12.210
perfectly well-defined
Fourier transform.
00:13:12.210 --> 00:13:15.310
Therefore, the inverse Fourier
transform of this nice looking
00:13:15.310 --> 00:13:18.590
thing is absolutely weird.
00:13:18.590 --> 00:13:20.840
The other reason is even
more important.
00:13:20.840 --> 00:13:23.490
We have to deal with
random processes.
00:13:23.490 --> 00:13:26.430
We have to deal with noise
functions, which are
00:13:26.430 --> 00:13:29.460
continuously varying
functions of time.
00:13:29.460 --> 00:13:32.660
We have to deal with what
we transmit, which is
00:13:32.660 --> 00:13:36.270
continuously varying
functions of time.
00:13:36.270 --> 00:13:39.530
Just like when we were dealing
with sources, we said if we
00:13:39.530 --> 00:13:44.250
want to compress a source it's
not enough to think about what
00:13:44.250 --> 00:13:46.110
one source sequence is.
00:13:46.110 --> 00:13:49.060
We have to think about
it probabalistically.
00:13:49.060 --> 00:13:55.260
Namely, we have to model these
functions as sample values of
00:13:55.260 --> 00:13:59.500
what we call a stochastic
process or a random process.
00:13:59.500 --> 00:14:02.150
In other words, when we start
talking about that, these
00:14:02.150 --> 00:14:06.040
functions which already look
complicated just become sample
00:14:06.040 --> 00:14:10.000
points in this much bigger space
where we're dealing with
00:14:10.000 --> 00:14:11.640
random processes.
00:14:11.640 --> 00:14:15.090
Now, when you're dealing with
sample points of these random
00:14:15.090 --> 00:14:20.460
processes, weirdness just crops
up as a necessary part
00:14:20.460 --> 00:14:21.830
of all of this.
00:14:21.830 --> 00:14:26.750
You can't define a random
process which consists of only
00:14:26.750 --> 00:14:28.540
non-weird things.
00:14:28.540 --> 00:14:31.430
If you do you get something that
doesn't work very well,
00:14:31.430 --> 00:14:33.350
it doesn't do much for you.
00:14:33.350 --> 00:14:37.200
People do that all the time, but
you run out of steam with
00:14:37.200 --> 00:14:39.250
it very, very quickly.
00:14:39.250 --> 00:14:42.440
So for both reasons we
have to be able to
00:14:42.440 --> 00:14:44.090
deal with weird things.
00:14:44.090 --> 00:14:47.940
The most ideal thing is to be
able to deal with weird things
00:14:47.940 --> 00:14:51.750
and get rid of them without
even thinking about them.
00:14:51.750 --> 00:14:54.890
What's the nice thing about
Lebesgue theory?
00:14:54.890 --> 00:14:57.270
It let's you get rid of all
the weirdness without even
00:14:57.270 --> 00:14:59.690
thinking about it.
00:14:59.690 --> 00:15:02.550
But in order to understand it to
start with so we get rid of
00:15:02.550 --> 00:15:06.440
all that weird stuff, we have
to understand just a little
00:15:06.440 --> 00:15:09.710
bit about what it's about
to start with.
00:15:09.710 --> 00:15:16.610
So that's where we're going.
00:15:16.610 --> 00:15:21.790
Well, in this picture --
let me put the picture
00:15:21.790 --> 00:15:26.420
back up again --
00:15:26.420 --> 00:15:28.250
I sort of sluffed
over something.
00:15:28.250 --> 00:15:31.850
For a nice, well-behaved
function, it's easy to find
00:15:31.850 --> 00:15:37.020
out what the measure
of a function is.
00:15:37.020 --> 00:15:41.560
Namely, what the measure is of
the set of values where you're
00:15:41.560 --> 00:15:45.270
in some tiny little interval,
and that's straightforward, it
00:15:45.270 --> 00:15:49.310
was just the sum of a
bunch of intervals.
00:15:52.290 --> 00:15:56.000
But now we come to the clincher,
which is we want to
00:15:56.000 --> 00:15:59.380
be able to deal with weird
functions too, and for weird
00:15:59.380 --> 00:16:05.540
functions this measure, the set
of values of time in which
00:16:05.540 --> 00:16:08.780
the function is in some tiny
little slice here, the measure
00:16:08.780 --> 00:16:11.960
of that is going to become
rather complicated.
00:16:11.960 --> 00:16:14.760
Therefore, we have to understand
how to find the
00:16:14.760 --> 00:16:17.550
measure of some pretty
weird sets.
00:16:17.550 --> 00:16:22.730
So, Lebesgue went on and said
OK, I'll define how to find
00:16:22.730 --> 00:16:25.160
measure of things.
00:16:25.160 --> 00:16:26.910
It's all very intuitive.
00:16:26.910 --> 00:16:31.530
If you have any real
numbers, a and b --
00:16:31.530 --> 00:16:33.990
if I have two real numbers I
might as well say one of them
00:16:33.990 --> 00:16:37.120
was less than or equal to the
other one, so a is less than
00:16:37.120 --> 00:16:38.620
or equal to b.
00:16:38.620 --> 00:16:40.990
I want to include minus
infinity and
00:16:40.990 --> 00:16:42.510
plus infinity here.
00:16:42.510 --> 00:16:46.620
When we say the set of real
numbers, we don't include
00:16:46.620 --> 00:16:49.650
minus infinity and
plus infinity.
00:16:49.650 --> 00:16:52.800
When we talk about the extended
set of real numbers,
00:16:52.800 --> 00:16:56.370
we doing include minus infinity
and plus infinity,
00:16:56.370 --> 00:16:58.780
Here for the most part when
we're dealing with measure
00:16:58.780 --> 00:17:01.990
theory, we really want to
include minus infinity and
00:17:01.990 --> 00:17:06.190
plus infinity also because they
make things easier to
00:17:06.190 --> 00:17:07.430
talk about.
00:17:07.430 --> 00:17:13.140
So what Lebesgue said is for any
interval the set of points
00:17:13.140 --> 00:17:17.910
which lie between a and b, and
he started out with the open
00:17:17.910 --> 00:17:21.290
interval -- namely the set of
points not including a and not
00:17:21.290 --> 00:17:25.690
including b, but including all
the real numbers in between --
00:17:25.690 --> 00:17:29.510
the measure of that is what we
said before, and none of you
00:17:29.510 --> 00:17:30.950
objected to it.
00:17:30.950 --> 00:17:33.950
The measure of an interval ought
to be the size of the
00:17:33.950 --> 00:17:37.900
interval, because after all,
all Lebesgue was doing was
00:17:37.900 --> 00:17:41.430
taking size and putting another
name on it because we
00:17:41.430 --> 00:17:44.930
all thought of it
as being size.
00:17:44.930 --> 00:17:49.890
So the measure of the interval
ab is b minus a.
00:17:49.890 --> 00:17:52.390
Now as engineers we all know
that it doesn't really make
00:17:52.390 --> 00:17:55.050
any difference whether you
include a or you don't include
00:17:55.050 --> 00:17:58.170
a when we're trying to find
the size of an interval.
00:17:58.170 --> 00:18:03.880
So, he said OK, the size of that
interval is b minus a,
00:18:03.880 --> 00:18:08.370
whether or not you include
either of the end points.
00:18:08.370 --> 00:18:12.060
Then he went on to say something
else, the measure of
00:18:12.060 --> 00:18:15.740
a countable union of disjoint
intervals is the sum of the
00:18:15.740 --> 00:18:17.540
measure of each interval.
00:18:22.340 --> 00:18:26.040
We already said when we were
trying to figure out what the
00:18:26.040 --> 00:18:31.020
Lebesgue integral was, that if
you had several intervals, the
00:18:31.020 --> 00:18:34.845
measure of the entire interval,
namely the measure
00:18:34.845 --> 00:18:39.330
of the union of those intervals
ought to be the sum
00:18:39.330 --> 00:18:41.990
of the measure of
each interval.
00:18:41.990 --> 00:18:45.890
So all he's adding here is
let's go on and go to the
00:18:45.890 --> 00:18:51.410
limit and talk about a countably
infinite number of
00:18:51.410 --> 00:18:55.100
intervals and use the
same definition.
00:18:55.100 --> 00:18:58.840
Now there's one nice thing about
that and what is it?
00:18:58.840 --> 00:19:06.560
If we take the sum of a
countable set of things, each
00:19:06.560 --> 00:19:09.110
of those things is
non-negative.
00:19:09.110 --> 00:19:13.300
So what happens when we take
the sum of a bunch of
00:19:13.300 --> 00:19:14.780
non-negative numbers?
00:19:19.170 --> 00:19:21.120
There are only two possibilities
when you take
00:19:21.120 --> 00:19:26.430
the sum of even a countable set
of non-negative numbers
00:19:26.430 --> 00:19:27.210
and what are they?
00:19:27.210 --> 00:19:31.980
AUDIENCE: Either to the limit
or it goes to infinity?
00:19:31.980 --> 00:19:36.040
PROFESSOR: It goes to a limit
or it goes to infinity, yes.
00:19:36.040 --> 00:19:39.090
Lebesgue said well, let's say if
it goes to infinity that's
00:19:39.090 --> 00:19:41.230
a limit also.
00:19:41.230 --> 00:19:43.620
So, in other words, there are
only two possibilities -- the
00:19:43.620 --> 00:19:46.590
sum is finite, you
can find the sum,
00:19:46.590 --> 00:19:47.990
and the sum if infinite.
00:19:47.990 --> 00:19:50.170
Nothing else can happen.
00:19:50.170 --> 00:19:52.760
So that's nice.
00:19:52.760 --> 00:19:55.090
Then Lebesgue said
one other thing.
00:19:55.090 --> 00:19:58.800
A bunch of mathematicians trying
to deal with this did
00:19:58.800 --> 00:20:03.020
different things when they were
trying to deal with very
00:20:03.020 --> 00:20:05.490
small sets.
00:20:05.490 --> 00:20:08.230
Some of them said well these
very small sets aren't
00:20:08.230 --> 00:20:12.980
measurable, others, and part of
Lebesgue's genius was that
00:20:12.980 --> 00:20:17.200
he said if you can take a set of
points and you can put them
00:20:17.200 --> 00:20:20.740
inside another set of points,
which doesn't amount to
00:20:20.740 --> 00:20:25.800
anything, then this smaller set
of points should have a
00:20:25.800 --> 00:20:27.830
measure which is less
than or equal to the
00:20:27.830 --> 00:20:30.980
bigger set of points.
00:20:30.980 --> 00:20:37.240
So that that says that any
subset of something that has
00:20:37.240 --> 00:20:40.860
zero measure also ought
to have zero measure.
00:20:40.860 --> 00:20:42.750
Now when we look at some
examples you'll see that
00:20:42.750 --> 00:20:46.030
that's not quite as obvious
as it seems.
00:20:46.030 --> 00:20:47.810
But anyway, he said that.
00:20:47.810 --> 00:20:52.360
An even nicer way to put that
is when you're talking about
00:20:52.360 --> 00:20:59.010
weird intervals, try to
cover that weird set
00:20:59.010 --> 00:21:03.200
with a bunch of intervals.
00:21:03.200 --> 00:21:06.180
If you can cover it with
a bunch of intervals in
00:21:06.180 --> 00:21:09.640
different ways and the measure
of the bunch of intervals, the
00:21:09.640 --> 00:21:13.550
countable set of intervals, is
you can make it arbitrarily
00:21:13.550 --> 00:21:18.060
small, then we also say that
this set has measure zero.
00:21:18.060 --> 00:21:23.750
So any time you have a set s
and you can cover it with
00:21:23.750 --> 00:21:27.780
intervals which have arbitrarily
small measure,
00:21:27.780 --> 00:21:30.300
namely we can make the measure
of those intervals as small as
00:21:30.300 --> 00:21:34.910
we want to make them, we say
bingo, s has zero measure.
00:21:34.910 --> 00:21:38.000
Now that's going to be very
nice, because it let's us get
00:21:38.000 --> 00:21:40.430
rid of an awful lot of things.
00:21:40.430 --> 00:21:45.690
Because any time we have some
set which has zero measure in
00:21:45.690 --> 00:21:48.760
this sense, when we look back
at what we did with the
00:21:48.760 --> 00:21:53.140
Lebesgue integral, it says
good, forget about it.
00:21:53.140 --> 00:21:55.700
If it has zero measure
it doesn't come into
00:21:55.700 --> 00:21:58.620
the integral at all.
00:21:58.620 --> 00:22:00.320
That's exactly the
thing we want.
00:22:00.320 --> 00:22:01.880
We'd like to get rid
of all that stuff.
00:22:04.620 --> 00:22:07.820
So if we're going to get rid of
it we have these sets which
00:22:07.820 --> 00:22:11.450
have measure zero, they don't
amount to anything, and we're
00:22:11.450 --> 00:22:15.270
not going to worry about them.
00:22:15.270 --> 00:22:18.930
Let's do an example.
00:22:18.930 --> 00:22:21.960
It's a famous example.
00:22:21.960 --> 00:22:25.750
What about the set of rationals
which lie between
00:22:25.750 --> 00:22:28.010
zero and 1?
00:22:28.010 --> 00:22:32.020
Well, rational numbers are
numbers where there's an
00:22:32.020 --> 00:22:36.790
integer numerator and an integer
denominator, and
00:22:36.790 --> 00:22:40.270
people have shown that there
are numbers which are
00:22:40.270 --> 00:22:43.060
approximated by rational
numbers but they're not
00:22:43.060 --> 00:22:46.000
rational numbers.
00:22:46.000 --> 00:22:49.210
You can take that set of
rational numbers and you can
00:22:49.210 --> 00:22:50.330
order them.
00:22:50.330 --> 00:22:54.960
The way I've ordered them here,
you can't order them in
00:22:54.960 --> 00:22:56.610
terms of how big they are.
00:22:56.610 --> 00:23:00.400
You can't start with the
smallest rational number which
00:23:00.400 --> 00:23:04.830
is greater than zero because
there isn't any such thing.
00:23:04.830 --> 00:23:10.170
Whatever rational number you
find which is greater than
00:23:10.170 --> 00:23:15.580
zero, I can take a half of it,
that's a rational number, and
00:23:15.580 --> 00:23:17.460
that also is greater
than zero and it's
00:23:17.460 --> 00:23:19.600
smaller than your number.
00:23:19.600 --> 00:23:21.840
If you don't like me getting
the better of you, you can
00:23:21.840 --> 00:23:24.770
then take half of that and come
up with a smaller number
00:23:24.770 --> 00:23:27.820
than I could find, and we can
go back and forth on this as
00:23:27.820 --> 00:23:30.530
long as we want.
00:23:30.530 --> 00:23:33.310
So the way we have to order
these is a little
00:23:33.310 --> 00:23:34.830
more subtle than that.
00:23:34.830 --> 00:23:38.780
Here we're going to order them
in terms of first the size of
00:23:38.780 --> 00:23:42.330
the denominator and next, the
size of the numerator.
00:23:42.330 --> 00:23:45.690
So there's only one fraction
in this interval --
00:23:50.980 --> 00:23:52.290
oh, I screwed that up.
00:23:52.290 --> 00:23:55.960
I really wanted to look at the
set of rational in the open
00:23:55.960 --> 00:24:00.123
interval between zero and 1,
because I don't want zero in
00:24:00.123 --> 00:24:03.240
it, and I don't want 1 in it,
but that's a triviality.
00:24:03.240 --> 00:24:06.350
You can put zero and
1 in or not.
00:24:06.350 --> 00:24:08.770
So, anyway, we'll leave
zero and 1 out.
00:24:08.770 --> 00:24:10.830
So I start out with 1/2 --
00:24:10.830 --> 00:24:14.890
that's the only rational number
strictly between zero
00:24:14.890 --> 00:24:17.560
and 1 with a denominator of 2.
00:24:17.560 --> 00:24:23.520
Then look at the numbers which
have a denominator of 3, so I
00:24:23.520 --> 00:24:25.830
have 1/3 and 2/3.
00:24:25.830 --> 00:24:29.570
I look then at the rational
numbers which have a
00:24:29.570 --> 00:24:31.130
denominator of 4.
00:24:31.130 --> 00:24:34.620
I have 1/4, I have to leave out
2/4, because that's the
00:24:34.620 --> 00:24:38.860
same as 1/2 which I've already
counted, so I have 3/4, Then I
00:24:38.860 --> 00:24:44.900
go on to 1/5, 2/5, 3/5,
4/5 and so forth.
00:24:44.900 --> 00:24:50.230
In doing this I have actually
counted what the integers are.
00:24:50.230 --> 00:24:53.600
Namely, whenever you can take
a set and put it into
00:24:53.600 --> 00:24:57.020
correspondence with the
integers, you're showing that
00:24:57.020 --> 00:24:58.840
it's countable.
00:24:58.840 --> 00:25:00.760
I've even labeled what
the counting is.
00:25:00.760 --> 00:25:05.190
A sub 1 is 1/2, A sub 2
is 1/3 and so forth.
00:25:05.190 --> 00:25:12.500
Now, I want to stick this set
inside of a set of intervals,
00:25:12.500 --> 00:25:14.220
and that's pretty easy.
00:25:14.220 --> 00:25:20.130
I stick A sub i inside of the
closed interval, which is A
00:25:20.130 --> 00:25:23.460
sub i on one side and A sub
i on the other side.
00:25:23.460 --> 00:25:25.820
So I'm sticking it inside
of an interval
00:25:25.820 --> 00:25:29.310
which has zero measure.
00:25:29.310 --> 00:25:32.010
Then what I'm going to do is
I'm going to add up all of
00:25:32.010 --> 00:25:32.560
those things.
00:25:32.560 --> 00:25:35.220
Whenever you add up an infinite
number of things, you
00:25:35.220 --> 00:25:38.240
really have to add up a finite
number of them and then look
00:25:38.240 --> 00:25:41.130
at what happens when you
go to the limit.
00:25:41.130 --> 00:25:46.980
So I add up all of these zeroes,
and when I add up n of
00:25:46.980 --> 00:25:48.570
them I get zero.
00:25:48.570 --> 00:25:52.460
I continue to add zeroes, I
continue to have zero, and the
00:25:52.460 --> 00:25:55.370
limit is zero.
00:25:55.370 --> 00:25:59.380
Now here's where the
mathematicians have pulled a
00:25:59.380 --> 00:26:04.140
very clever swindle on you,
because what they're saying is
00:26:04.140 --> 00:26:07.640
that infinity times
zero is zero.
00:26:07.640 --> 00:26:12.510
But they're saying it in
a very precise way.
00:26:12.510 --> 00:26:16.760
But anyway, since we said it
in a very precise way,
00:26:16.760 --> 00:26:21.740
infinity times zero here is
zero, the way we've said it.
00:26:21.740 --> 00:26:25.640
But somehow that's not
very satisfying.
00:26:25.640 --> 00:26:28.990
I mean it really looks
like we've cheated.
00:26:28.990 --> 00:26:31.320
So let's go on and do this
in a different way.
00:26:34.410 --> 00:26:37.670
What we're going to do now is
for each of these rational
00:26:37.670 --> 00:26:41.330
numbers, we're going to put a
little hat around them, a
00:26:41.330 --> 00:26:43.640
little rectangular
hat around them.
00:26:43.640 --> 00:26:50.150
Namely a little interval which
includes A sub i, and which
00:26:50.150 --> 00:26:55.040
also goes delta over 2 that way,
delta over 2 this way,
00:26:55.040 --> 00:26:57.650
and also multiplies that by --
00:27:07.270 --> 00:27:10.500
my computer knew I was going
to talk about computers
00:27:10.500 --> 00:27:13.360
replacing people, so it
made some mistakes to
00:27:13.360 --> 00:27:15.160
get even with me.
00:27:15.160 --> 00:27:19.210
So this is delta times 2 to the
minus i minus 1, and delta
00:27:19.210 --> 00:27:22.030
times 2 to the minus
i minus 1.
00:27:22.030 --> 00:27:26.080
In other words, you take 1/2
and you put a pretty big
00:27:26.080 --> 00:27:27.330
interval around it.
00:27:27.330 --> 00:27:32.390
You take 1/3, you put a smaller
interval around it.
00:27:32.390 --> 00:27:37.480
2/3 you put a smaller interval
around that and so forth.
00:27:37.480 --> 00:27:41.670
Well, these intervals here
are going to overlap.
00:27:41.670 --> 00:27:46.290
But anyway, the union of two
overlapping open intervals is
00:27:46.290 --> 00:27:49.230
an open interval which
has a smaller measure
00:27:49.230 --> 00:27:50.120
than with two of them.
00:27:50.120 --> 00:27:54.980
In other words, if you take
this, the measure of this and
00:27:54.980 --> 00:27:57.990
add it to the measure of this,
the union of this and
00:27:57.990 --> 00:28:00.030
this is just this.
00:28:03.630 --> 00:28:07.020
We don't have to be very
sophisticated to see that this
00:28:07.020 --> 00:28:12.800
length is less than this length
plus length, because
00:28:12.800 --> 00:28:15.750
I've double counted
things here.
00:28:15.750 --> 00:28:19.070
So anyway, when I do this I add
up the measure of all of
00:28:19.070 --> 00:28:23.140
these things and I get delta,
and I then let delta get as
00:28:23.140 --> 00:28:25.500
small as I want to.
00:28:25.500 --> 00:28:29.470
Again, I find that the measure
of the rationals is zero.
00:28:33.840 --> 00:28:38.990
Now if you don't like this you
should be very happy with
00:28:38.990 --> 00:28:44.530
yourselves, because I've
struggled with this for years
00:28:44.530 --> 00:28:48.780
and it's not intuitive, because
with these intervals
00:28:48.780 --> 00:28:51.910
I'm putting in here, no matter
how small I make that
00:28:51.910 --> 00:28:55.600
interval, there's an infinite
number of rational numbers
00:28:55.600 --> 00:28:58.560
which are in that interval.
00:28:58.560 --> 00:29:00.820
In other words, the thing
we're trying to do is to
00:29:00.820 --> 00:29:10.420
separate the interval 0,1 into
some union of intervals which
00:29:10.420 --> 00:29:15.400
don't amount to anything but
which include all of the
00:29:15.400 --> 00:29:17.660
rational numbers.
00:29:17.660 --> 00:29:21.700
Somehow this argument is a
little bit bogus because no
00:29:21.700 --> 00:29:25.620
matter what number I look at
between zero and 1, there are
00:29:25.620 --> 00:29:30.250
rational numbers arbitrarily
close to it.
00:29:30.250 --> 00:29:35.030
In other words, what's going on
here is strictly a matter
00:29:35.030 --> 00:29:39.010
of which order we take limits
in, and that's what makes the
00:29:39.010 --> 00:29:42.160
argument subtle.
00:29:42.160 --> 00:29:44.920
But anyway, that is
a perfectly sound
00:29:44.920 --> 00:29:46.130
mathematical argument.
00:29:46.130 --> 00:29:48.060
You can't get around it.
00:29:48.060 --> 00:29:50.760
It's why people objected to what
Lebesgue was doing for a
00:29:50.760 --> 00:29:54.370
long time, because it wasn't
intuitive to them either.
00:29:54.370 --> 00:29:59.260
It was intuitive to Lebesgue,
and finally it's become
00:29:59.260 --> 00:30:03.080
intuitive to everyone, but
not really intuitive.
00:30:03.080 --> 00:30:06.970
It's just that mathematicians
have heard it so many times
00:30:06.970 --> 00:30:09.380
that they believe it.
00:30:09.380 --> 00:30:13.360
I mean one of the problems with
any society is that if
00:30:13.360 --> 00:30:15.810
you tell people things
often enough they
00:30:15.810 --> 00:30:18.760
start to believe them.
00:30:18.760 --> 00:30:21.720
Unfortunately, that's true
in mathematics, too.
00:30:21.720 --> 00:30:25.250
Fortunately in mathematics we
have proofs of things, so that
00:30:25.250 --> 00:30:27.860
when somebody is telling you
something again and again
00:30:27.860 --> 00:30:30.780
which is false, there are always
people who will look at
00:30:30.780 --> 00:30:33.670
it and say no, that's
not true.
00:30:33.670 --> 00:30:37.500
Whereas in other cases,
not necessarily.
00:30:40.230 --> 00:30:45.300
There are also uncountable
sets with measure zero.
00:30:45.300 --> 00:30:48.260
For those of you who are already
sort of overwhelmed by
00:30:48.260 --> 00:30:51.890
this, why don't you go to sleep
for three minutes, it'll
00:30:51.890 --> 00:30:54.960
only take three minutes to
talk about this, but
00:30:54.960 --> 00:30:58.450
it's kind of cute.
00:30:58.450 --> 00:31:01.920
In the set I want to talk
about something closely
00:31:01.920 --> 00:31:06.990
related to what people call
the Cantor set, but it's a
00:31:06.990 --> 00:31:08.590
little bit simpler than that.
00:31:08.590 --> 00:31:11.220
So what I'd like to
do, and this is
00:31:11.220 --> 00:31:13.350
already familiar to you.
00:31:13.350 --> 00:31:17.850
I can take numbers between zero
and 1 and I can represent
00:31:17.850 --> 00:31:19.460
them in a binary expansion.
00:31:19.460 --> 00:31:22.820
I can also represent them
in a ternary expansion.
00:31:22.820 --> 00:31:26.140
I can also represent them in a
decimal expansion, which is
00:31:26.140 --> 00:31:28.770
what you've been doing since
you were three years old or
00:31:28.770 --> 00:31:32.110
five years older or whenever
you started doing this.
00:31:32.110 --> 00:31:35.500
Well, ternary is simpler than
decimal, so you could have
00:31:35.500 --> 00:31:38.960
done this a year before you
started to deal with decimal
00:31:38.960 --> 00:31:41.590
expansions.
00:31:41.590 --> 00:31:45.170
So I want to look at all of
the ternary expansions.
00:31:45.170 --> 00:31:49.870
Each real number corresponds to
a ternary expansion, which
00:31:49.870 --> 00:31:55.480
is an infinite sequence of
numbers each of which are
00:31:55.480 --> 00:31:56.910
zero, 1 or 2.
00:31:59.880 --> 00:32:04.170
Now, what I'm going to do is I'm
going to remove all of the
00:32:04.170 --> 00:32:09.750
sequences which contain any
1's in them at all.
00:32:09.750 --> 00:32:13.510
Now it's not immediately clear
what that's going to do, but
00:32:13.510 --> 00:32:15.150
think of it this way.
00:32:15.150 --> 00:32:18.790
If I first look at the
sequences, I'm going to remove
00:32:18.790 --> 00:32:21.340
all the sequences which
start with 1.
00:32:21.340 --> 00:32:24.330
So the sequences which start
with 1 and have anything else
00:32:24.330 --> 00:32:29.820
after them, that's really the
interval that starts at 1/3
00:32:29.820 --> 00:32:34.760
and ends at 2/3, because that's
really what starting
00:32:34.760 --> 00:32:36.510
with 1 means.
00:32:36.510 --> 00:32:38.130
This is what we talked about
when we talked about
00:32:38.130 --> 00:32:41.260
approximating binary numbers
also, if you remember, is the
00:32:41.260 --> 00:32:43.790
way we proved the Kraft
inequality.
00:32:43.790 --> 00:32:46.985
It was the same idea.
00:32:46.985 --> 00:32:53.800
The sequences which have a 1 in
the second position, when
00:32:53.800 --> 00:32:59.180
we remove them we're removing
the interval from 1/9 to 2/9.
00:32:59.180 --> 00:33:03.390
We're also removing the interval
from 7/9 to 8/9.
00:33:03.390 --> 00:33:07.240
We're also removing the 4/9 to
5/9 interval, but we removed
00:33:07.240 --> 00:33:08.650
that before.
00:33:08.650 --> 00:33:17.130
So we wind up with something
which is now -- we've taken
00:33:17.130 --> 00:33:20.990
out this, we've now taken
out this, and
00:33:20.990 --> 00:33:22.200
we've taken out this.
00:33:22.200 --> 00:33:29.610
So the only thing left
is this and this and
00:33:29.610 --> 00:33:33.130
this and this, right?
00:33:33.130 --> 00:33:35.620
And we've removed
everything else.
00:33:35.620 --> 00:33:38.390
We keep on doing this.
00:33:38.390 --> 00:33:44.110
Well, each time we remove one
of these, all of the numbers
00:33:44.110 --> 00:33:50.280
that's -- when I do this n
times, the first time I go
00:33:50.280 --> 00:33:54.340
through this process, I removed
1/3 of the numbers,
00:33:54.340 --> 00:33:58.300
I'm left with 2/3
of the interval.
00:33:58.300 --> 00:34:02.010
When I remove everything that
has a 1 in the second
00:34:02.010 --> 00:34:06.840
position, I'm down
with 2/3 squared.
00:34:06.840 --> 00:34:11.540
When I remove everything which
has a 1 in the third position,
00:34:11.540 --> 00:34:14.370
I'm down to 2/3 cubed.
00:34:14.370 --> 00:34:16.480
When I keep on doing this
forever, what happens
00:34:16.480 --> 00:34:18.820
to 2/3 to the n?
00:34:18.820 --> 00:34:22.990
Well, 2/3 to the
n goes to zero.
00:34:22.990 --> 00:34:28.370
In other words, I have removed
an interval, I have removed a
00:34:28.370 --> 00:34:32.030
set, the measure 1, and
therefore I'm left with a set
00:34:32.030 --> 00:34:33.440
of measure zero.
00:34:33.440 --> 00:34:35.290
You can see this happening.
00:34:35.290 --> 00:34:40.220
I mean you only have diddly left
here and I keep cutting
00:34:40.220 --> 00:34:41.910
away at it.
00:34:41.910 --> 00:34:45.600
So less and less gets left.
00:34:45.600 --> 00:34:52.040
But what we now have is all
sequences, all infinite
00:34:52.040 --> 00:34:57.700
sequences of zeroes and 2's.
00:34:57.700 --> 00:35:01.760
So I'm left with all binary
sequences except instead of
00:35:01.760 --> 00:35:05.480
binary sequences with zeros
and 1's, I now have binary
00:35:05.480 --> 00:35:08.880
sequences with zeros and 2's.
00:35:08.880 --> 00:35:16.150
How many binary sequences are
there when I continue forever?
00:35:16.150 --> 00:35:19.070
Well, you know they're an
uncountable number, because if
00:35:19.070 --> 00:35:22.810
I take all the numbers between
zero and 1, I represent them
00:35:22.810 --> 00:35:25.460
in binary zeroes and
1's, I have an
00:35:25.460 --> 00:35:29.540
uncountable number of them.
00:35:29.540 --> 00:35:33.480
Well, because I have to have an
uncountable number because
00:35:33.480 --> 00:35:36.710
we already showed that any
countable set doesn't amount
00:35:36.710 --> 00:35:38.230
to anything.
00:35:38.230 --> 00:35:41.770
Countable sets are diddly.
00:35:41.770 --> 00:35:44.270
Countable sets all
just go away.
00:35:44.270 --> 00:35:48.430
So, anything which gets left
has to be uncountable.
00:35:48.430 --> 00:35:51.260
Again, people had to worry about
this for a long time.
00:35:51.260 --> 00:35:55.380
But anyway, this gives you an
uncountable set which has
00:35:55.380 --> 00:35:56.630
measure zero.
00:35:59.410 --> 00:36:02.360
So, back to measurable
functions.
00:36:02.360 --> 00:36:05.830
I'm going to get off of
mathematics relatively soon,
00:36:05.830 --> 00:36:08.730
but we need at least this
much to figure out
00:36:08.730 --> 00:36:10.030
what's going on here.
00:36:10.030 --> 00:36:12.250
We say that a function
is measurable.
00:36:12.250 --> 00:36:15.740
Before we were only talking
about sets of numbers being
00:36:15.740 --> 00:36:18.240
measurable.
00:36:18.240 --> 00:36:21.950
We had to talk about sets of
numbers being measurable
00:36:21.950 --> 00:36:26.330
because we were interested in
the question of what's the set
00:36:26.330 --> 00:36:32.440
of times for which a function
lies between 2 epsilon and 3
00:36:32.440 --> 00:36:35.370
epsilon, for example.
00:36:35.370 --> 00:36:39.530
What we said is we can say a
great deal about that because
00:36:39.530 --> 00:36:44.120
we can not only add up a bunch
of intervals, we can also add
00:36:44.120 --> 00:36:47.100
up a countable bunch of
intervals, and we can also get
00:36:47.100 --> 00:36:50.120
rid of anything which
is negligible.
00:36:50.120 --> 00:36:56.220
So, a function is measurable if
the set of t, such that u
00:36:56.220 --> 00:36:58.970
of t lies between these
two points is
00:36:58.970 --> 00:37:00.870
measurable for each interval.
00:37:00.870 --> 00:37:05.230
In other words, if no matter how
I split up this interval,
00:37:05.230 --> 00:37:13.660
if no matter what slice I look
at, the set of times over
00:37:13.660 --> 00:37:17.820
which the function lies in
there is measurable.
00:37:17.820 --> 00:37:21.010
That's what a measurable
function is.
00:37:21.010 --> 00:37:25.000
Everybody understand
what I just said?
00:37:25.000 --> 00:37:27.520
Let me try to say
it once more.
00:37:27.520 --> 00:37:33.300
A function is measurable if for
every two values, say 3
00:37:33.300 --> 00:37:38.840
epsilon and 2 epsilon, if the
set of values t for which the
00:37:38.840 --> 00:37:43.340
function lies between 2 epsilon
and 3 epsilon, if that
00:37:43.340 --> 00:37:46.030
set is measurable.
00:37:46.030 --> 00:37:49.550
In other words, that's the set
we were talking about before
00:37:49.550 --> 00:37:53.640
which went from here to
here, and which went
00:37:53.640 --> 00:37:55.660
from here to there.
00:37:55.660 --> 00:37:58.700
In this case for this very
simple function, that's just
00:37:58.700 --> 00:38:00.520
the sum of two intervals.
00:38:00.520 --> 00:38:04.370
If I make the function wiggle
a great deal more, it's the
00:38:04.370 --> 00:38:07.850
sum of a lot more intervals.
00:38:07.850 --> 00:38:12.075
So, we say the function is
measurable if all of the sets
00:38:12.075 --> 00:38:13.560
are measurable.
00:38:13.560 --> 00:38:16.900
Now, what I'm going to do is
when I'm trying to define this
00:38:16.900 --> 00:38:19.200
integral, I'm going
to have to go to
00:38:19.200 --> 00:38:22.450
smaller and smaller intervals.
00:38:22.450 --> 00:38:25.380
Let's start out with epsilon,
2 epsilon, 3
00:38:25.380 --> 00:38:27.240
epsilon and so forth.
00:38:27.240 --> 00:38:29.360
Let's look at a non-negative
function--.
00:38:31.970 --> 00:38:32.640
Yeah?
00:38:32.640 --> 00:38:35.928
AUDIENCE: Maybe I missed
something, but could you tell
00:38:35.928 --> 00:38:40.300
me [UNINTELLIGIBLE] definition
for a set, if measurable.
00:38:40.300 --> 00:38:41.870
PROFESSOR: What's the definition
for a set is
00:38:41.870 --> 00:38:44.710
measurable.
00:38:44.710 --> 00:38:49.350
I didn't really say,
and that's good.
00:38:49.350 --> 00:38:51.650
I gave you a bunch of conditions
under which a set
00:38:51.650 --> 00:38:55.560
is measurable, and if I have
enough conditions for which
00:38:55.560 --> 00:38:59.710
it's measurable then I don't
have to worry about--.
00:39:03.060 --> 00:39:07.010
I said that it is measurable
under all of these conditions.
00:39:07.010 --> 00:39:09.460
I'm saying I don't have to worry
about the rest of them
00:39:09.460 --> 00:39:12.790
because these are enough
conditions to talk about
00:39:12.790 --> 00:39:14.180
everything I want
to talk about.
00:39:14.180 --> 00:39:15.430
AUDIENCE: [UNINTELLIGIBLE].
00:39:17.880 --> 00:39:19.350
PROFESSOR: I will define
my measure as
00:39:19.350 --> 00:39:21.360
all of these things.
00:39:21.360 --> 00:39:23.540
Unfortunately, you need a little
bit more, and if you
00:39:23.540 --> 00:39:26.750
want to get more you better
take a course in real
00:39:26.750 --> 00:39:31.640
variables and measure theory.
00:39:31.640 --> 00:39:32.890
Good.
00:39:38.730 --> 00:39:46.960
So, if I want to now make this
epsilon smaller, what I'm
00:39:46.960 --> 00:39:48.980
going to do is do it in
a particular way.
00:39:48.980 --> 00:39:51.920
I'm going to start out
partitioning this into
00:39:51.920 --> 00:39:54.270
intervals of size epsilon.
00:39:54.270 --> 00:39:57.800
Then I'm going to partition
it into intervals of size
00:39:57.800 --> 00:39:59.700
epsilon over 2.
00:39:59.700 --> 00:40:04.150
When I partition it into
intervals of size epsilon over
00:40:04.150 --> 00:40:07.970
2, I'm adding a bunch
of extra things.
00:40:07.970 --> 00:40:13.580
This thing gets added because
when I'm looking at the
00:40:13.580 --> 00:40:20.380
interval between epsilon and 3
epsilon over 2, the function
00:40:20.380 --> 00:40:29.100
is in this interval here, over
this [UNINTELLIGIBLE].
00:40:32.110 --> 00:40:37.990
It's in this interval over
this whole thing.
00:40:37.990 --> 00:40:42.840
I'm representing it by
this value down here.
00:40:42.840 --> 00:40:48.030
Now when I have this tinier
interval, I see that this
00:40:48.030 --> 00:40:52.530
function is really in this
interval from here to there
00:40:52.530 --> 00:40:57.050
also, and therefore, instead of
representing the function
00:40:57.050 --> 00:41:01.210
over this interval by epsilon,
I'm representing it by 3
00:41:01.210 --> 00:41:03.060
epsilon over 2.
00:41:03.060 --> 00:41:07.540
In other words, as I add these
extra quantization levels, I
00:41:07.540 --> 00:41:11.850
can never lose anything,
I only gain things.
00:41:11.850 --> 00:41:15.370
So I gain all of these
cross-hatched regions when I
00:41:15.370 --> 00:41:20.260
do this, which says that when
I add up all these things in
00:41:20.260 --> 00:41:25.960
the integral, every time I
decrease epsilon by 2, the
00:41:25.960 --> 00:41:28.520
integral that I've got,
the approximation to
00:41:28.520 --> 00:41:30.550
the integral increases.
00:41:30.550 --> 00:41:35.960
Now what happens when you take
a sum of a set of numbers
00:41:35.960 --> 00:41:38.160
which are increasing?
00:41:38.160 --> 00:41:40.870
Well, they're increasing, the
result that you get when you
00:41:40.870 --> 00:41:46.200
add them all up is increasing
also, and therefore, as I go
00:41:46.200 --> 00:41:50.090
from epsilon to epsilon over
2 to epsilon over 4 and so
00:41:50.090 --> 00:41:53.130
forth, I keep climbing up.
00:41:53.130 --> 00:41:56.540
Conclusion, I either get to a
finite number or I get to
00:41:56.540 --> 00:42:00.030
infinity -- only two
possibilities.
00:42:00.030 --> 00:42:04.820
Which says that if I'm looking
at non-negative functions, if
00:42:04.820 --> 00:42:08.570
I'm only looking at real
functions which have
00:42:08.570 --> 00:42:13.070
non-negative values, the
Lebesgue integral for a
00:42:13.070 --> 00:42:18.310
measurable function always
exists, if I include infinite
00:42:18.310 --> 00:42:20.210
limits as well as
finite limits.
00:42:23.860 --> 00:42:27.600
Now, if you think back to what
you learned about integration,
00:42:27.600 --> 00:42:30.980
and I hope you at least learned
enough about it that
00:42:30.980 --> 00:42:34.390
you remember there are a lot of
very nasty conditions about
00:42:34.390 --> 00:42:38.950
when integrals exist and
when they don't exist.
00:42:38.950 --> 00:42:40.640
Here that's all gone away.
00:42:43.230 --> 00:42:46.080
This is a beautifully
simple statement.
00:42:46.080 --> 00:42:50.040
You take the integral of a
non-negative function, if it's
00:42:50.040 --> 00:42:54.140
measurable, and there are
only two possibilities.
00:42:54.140 --> 00:42:57.110
The integral of some
finite number or
00:42:57.110 --> 00:42:59.340
the integral is infinite.
00:42:59.340 --> 00:43:01.950
It's never undefined,
it's always defined.
00:43:06.010 --> 00:43:08.480
I think that's neat.
00:43:08.480 --> 00:43:12.680
A few people don't think
it's neat, too bad.
00:43:17.100 --> 00:43:19.760
I guess when I was first
studying this, I didn't think
00:43:19.760 --> 00:43:22.660
it was neat either because
it was too complicated.
00:43:22.660 --> 00:43:26.450
So you have an excuse.
00:43:26.450 --> 00:43:29.030
If you think about it for a
while and you understand it
00:43:29.030 --> 00:43:31.040
and you don't think it's
neat, then I think
00:43:31.040 --> 00:43:32.290
you have a real problem.
00:43:36.090 --> 00:43:37.570
So now --
00:43:40.590 --> 00:43:41.840
I did this.
00:43:45.926 --> 00:43:48.070
I'm getting too many
slides out here.
00:43:51.970 --> 00:43:53.520
Here we go.
00:43:53.520 --> 00:43:55.860
Here's something new.
00:43:55.860 --> 00:43:57.790
Hardly looks new.
00:43:57.790 --> 00:44:02.220
Let's look at a function now,
just defined on the
00:44:02.220 --> 00:44:05.430
interval zero to 1.
00:44:05.430 --> 00:44:11.420
Suppose that h of t is equal to
1 for each rational number
00:44:11.420 --> 00:44:16.510
and at zero for each
irrational number.
00:44:16.510 --> 00:44:22.150
In other words, this is
a function which looks
00:44:22.150 --> 00:44:23.640
absolutely wild.
00:44:26.190 --> 00:44:35.100
It just goes up to here
and it's 1 or zero.
00:44:35.100 --> 00:44:39.590
It's 1 at this dense set of
points, which we've already
00:44:39.590 --> 00:44:42.460
said doesn't amount to anything,
and it's zero
00:44:42.460 --> 00:44:45.790
everywhere else.
00:44:45.790 --> 00:44:49.160
Now you put that into the
Reimann integral, and the
00:44:49.160 --> 00:44:53.080
Reimann integral goes crazy,
because no matter how small I
00:44:53.080 --> 00:44:56.820
make this interval, there are an
infinite number of rational
00:44:56.820 --> 00:45:00.040
numbers in that interval, and
therefore, the Reimann
00:45:00.040 --> 00:45:03.090
integral can never
even get started.
00:45:03.090 --> 00:45:06.780
For the Lebesgue integral, on
the other hand, look at what
00:45:06.780 --> 00:45:07.940
happens now.
00:45:07.940 --> 00:45:12.040
We have a bunch of points which
are sitting at 1, we
00:45:12.040 --> 00:45:14.840
have a bunch of points which
are sitting at zero.
00:45:14.840 --> 00:45:18.750
The only thing we have to do is
evaluate the measure of the
00:45:18.750 --> 00:45:23.090
set of points which are up in
some tiny interval up in here.
00:45:26.000 --> 00:45:30.360
What's this measure of the set
of t's corresponding to the
00:45:30.360 --> 00:45:32.450
rational numbers?
00:45:32.450 --> 00:45:36.350
Well, you already said
that was zero.
00:45:36.350 --> 00:45:40.490
Now, that's why Lebesgue
integration works.
00:45:40.490 --> 00:45:45.260
Any countable set, and in fact,
any of these uncountable
00:45:45.260 --> 00:45:51.110
sets that measure zero get lost
in here because you're
00:45:51.110 --> 00:45:54.610
combining them all together
and you say they don't
00:45:54.610 --> 00:45:58.490
contribute to the
integral at all.
00:45:58.490 --> 00:46:01.660
That's why Lebesgue integration
is so simple.
00:46:01.660 --> 00:46:03.530
You can forget about
all that stuff.
00:46:08.410 --> 00:46:13.830
When we looked last time at
the Fourier series for a
00:46:13.830 --> 00:46:18.960
square wave, you remember we
found that everything behaved
00:46:18.960 --> 00:46:22.690
very nicely, except where
the square wave had a
00:46:22.690 --> 00:46:26.350
discontinuity, the Fourier
series converged to the
00:46:26.350 --> 00:46:31.430
mid-point, and that was
kind of awkward.
00:46:31.430 --> 00:46:36.550
Well, the mid-points where the
function is discontinuous
00:46:36.550 --> 00:46:40.530
don't amount to anything,
because in that case there
00:46:40.530 --> 00:46:43.160
were just two of them, there
were only two points.
00:46:43.160 --> 00:46:47.870
If they had measure zero
it just washes away.
00:46:47.870 --> 00:46:51.030
You all felt intuitively when
you saw that example, that
00:46:51.030 --> 00:46:53.690
those points were
not important.
00:46:53.690 --> 00:46:57.370
You felt that this was
mathematical carping.
00:46:57.370 --> 00:47:00.600
Well, Lebesgue felt it was
mathematical carping too, but
00:47:00.600 --> 00:47:02.900
he went one step further and
he said here's a way of
00:47:02.900 --> 00:47:04.710
getting rid of all of that
and not having to
00:47:04.710 --> 00:47:06.540
worry about it anymore.
00:47:06.540 --> 00:47:09.250
So you've now gotten to the
point where you don't have to
00:47:09.250 --> 00:47:14.660
worry about any of this
stuff anymore.
00:47:20.920 --> 00:47:23.990
Now let's go a little
bit further.
00:47:23.990 --> 00:47:26.730
We're almost at the
end of this.
00:47:26.730 --> 00:47:31.640
If I take a function which maps
the real numbers into the
00:47:31.640 --> 00:47:34.620
real numbers, in other words,
it's a function which you can
00:47:34.620 --> 00:47:37.690
draw on the line.
00:47:37.690 --> 00:47:41.120
You take time going from minus
infinity to plus infinity, you
00:47:41.120 --> 00:47:44.310
define what this function
is at each time.
00:47:44.310 --> 00:47:46.910
That's what I'm talking
about here, a function
00:47:46.910 --> 00:47:48.160
which you can draw.
00:47:51.150 --> 00:47:56.540
The functions magnitude of
u of t, and the function
00:47:56.540 --> 00:48:03.910
magnitude of u of t squared
are both non-negative.
00:48:03.910 --> 00:48:07.620
Now I'm not going to prove this,
but it turns out that
00:48:07.620 --> 00:48:10.730
the magnitude and the magnitude
squared are both
00:48:10.730 --> 00:48:14.350
measurable functions if u of
t is a measurable function.
00:48:14.350 --> 00:48:17.190
In fact, from now on we're
just going to assume that
00:48:17.190 --> 00:48:21.270
everything we deal with is
measurable, every function is
00:48:21.270 --> 00:48:22.760
measurable.
00:48:22.760 --> 00:48:25.890
I challenge any of you without
looking it up in a book to
00:48:25.890 --> 00:48:28.570
find an example of a
non-measurable function.
00:48:28.570 --> 00:48:30.800
I challenge any of you to
find an example of a
00:48:30.800 --> 00:48:33.830
non-measurable set.
00:48:33.830 --> 00:48:37.470
I challenge any of you to
understand the definition of a
00:48:37.470 --> 00:48:40.590
non-measurable set if you
look it up in a book.
00:48:40.590 --> 00:48:43.310
You've heard about things like
the axiom of choice and things
00:48:43.310 --> 00:48:46.300
like that, which are very
fishy kinds of things --
00:48:46.300 --> 00:48:49.330
that's all involved in finding
non-measurable function.
00:48:49.330 --> 00:48:53.150
So, any function that you think
about is going to be
00:48:53.150 --> 00:48:56.300
measurable.
00:48:56.300 --> 00:49:00.080
I hate people who say things
like that, but it's the only
00:49:00.080 --> 00:49:02.250
way to get around this because
I don't want to give you any
00:49:02.250 --> 00:49:06.780
examples of that because
they're awful.
00:49:06.780 --> 00:49:12.890
Since magnitude of u of t and
magnitude of u of t squared
00:49:12.890 --> 00:49:16.220
are measurable and they're
non-negative,
00:49:16.220 --> 00:49:18.750
their integrals exist.
00:49:18.750 --> 00:49:21.780
Their integrals exist and are
either a finite number or
00:49:21.780 --> 00:49:24.040
they're infinite.
00:49:24.040 --> 00:49:29.710
So, we define L1 functions, and
we'll be dealing with L1
00:49:29.710 --> 00:49:33.910
functions and L2 functions all
the way through the course, u
00:49:33.910 --> 00:49:38.820
of t is an L1 function if it's
measurable, and if this
00:49:38.820 --> 00:49:41.760
integrals is less
than infinity.
00:49:41.760 --> 00:49:43.010
That's all there is to it.
00:49:45.440 --> 00:49:51.560
u2 is L2 if it's measurable
and the integral of u of t
00:49:51.560 --> 00:49:55.290
squared is less than infinity.
00:49:55.290 --> 00:49:59.120
I could have said that at the
beginning, but now you see
00:49:59.120 --> 00:50:03.040
that it makes a lot more sense
than it did before because we
00:50:03.040 --> 00:50:06.810
know that if u of t is
measurable, this integral
00:50:06.810 --> 00:50:10.990
exists -- it's either a finite
number or infinity.
00:50:10.990 --> 00:50:14.920
The L1 functions are those
particular functions where
00:50:14.920 --> 00:50:17.370
it's finite and not infinite.
00:50:17.370 --> 00:50:18.630
Same thing here.
00:50:18.630 --> 00:50:23.350
The L2 functions are those
where this is finite.
00:50:23.350 --> 00:50:27.320
This is really the energy of the
function, but now we can
00:50:27.320 --> 00:50:31.960
measure the energy of even weird
functions which are zero
00:50:31.960 --> 00:50:37.250
on the irrationals and
one on the rationals.
00:50:37.250 --> 00:50:41.840
Even things which are zero on
the non-cantor set points and
00:50:41.840 --> 00:50:46.710
1 on the cantor set points,
still it all works.
00:50:46.710 --> 00:50:53.850
So those define the
set L1 and L2.
00:50:53.850 --> 00:50:59.560
Now, a complex function u of
t, which maps r into c, why
00:50:59.560 --> 00:51:03.820
does a complex function
map r into c?
00:51:03.820 --> 00:51:05.180
What's the r doing there?
00:51:10.470 --> 00:51:15.180
Think of any old complex
function you can think of.
00:51:15.180 --> 00:51:18.155
e to the i, 2 pi t.
00:51:18.155 --> 00:51:22.410
That's something that wiggles
around, the sinusoid.
00:51:22.410 --> 00:51:25.010
t is a real number.
00:51:25.010 --> 00:51:36.510
So that function e to the i 2
pi t is mapping real numbers
00:51:36.510 --> 00:51:39.120
into complex numbers.
00:51:39.120 --> 00:51:41.910
That's what we mean by something
which maps the real
00:51:41.910 --> 00:51:44.130
numbers into complex numbers.
00:51:44.130 --> 00:51:46.480
We always call these
complex functions.
00:51:49.120 --> 00:51:51.430
I mean mathematicians
would say yeah, a
00:51:51.430 --> 00:51:53.470
function could be anything.
00:51:53.470 --> 00:51:56.700
But you know, when most of us
think of a function, we're
00:51:56.700 --> 00:52:01.060
thinking of mapping a real
variable into something else,
00:52:01.060 --> 00:52:03.230
and when we're thinking of
mapping a real variable into
00:52:03.230 --> 00:52:06.970
something else, we're usually
thinking of mapping it into
00:52:06.970 --> 00:52:10.610
real numbers or mapping it
into complex numbers, and
00:52:10.610 --> 00:52:13.950
because we want to deal with
these complex sinusoids, we
00:52:13.950 --> 00:52:16.320
have to include complex
numbers also.
00:52:19.170 --> 00:52:23.620
So a complex function is
measurable by definition if
00:52:23.620 --> 00:52:27.170
the real part and the imaginary
part are each
00:52:27.170 --> 00:52:28.480
measurable.
00:52:28.480 --> 00:52:33.300
We already know when a real
function is measurable.
00:52:33.300 --> 00:52:35.820
Namely, a real function is
measurable if each of these
00:52:35.820 --> 00:52:37.550
slices are measurable.
00:52:37.550 --> 00:52:41.470
So now we know when a complex
function is measurable.
00:52:41.470 --> 00:52:44.490
We already said that all of the
complex functions you can
00:52:44.490 --> 00:52:47.480
think of and all the ones we'll
ever deal with are all
00:52:47.480 --> 00:52:48.970
measurable.
00:52:48.970 --> 00:52:52.540
So L1 and L2 are defined in
the same way when we're
00:52:52.540 --> 00:52:55.020
dealing with complex
functions.
00:52:55.020 --> 00:52:58.390
Namely, just whether this
integral is less than infinity
00:52:58.390 --> 00:53:01.230
and this integral is
less than infinity.
00:53:03.730 --> 00:53:09.410
Since these functions -- this
is a real function from real
00:53:09.410 --> 00:53:12.660
into real, this is a real
function from real into real.
00:53:12.660 --> 00:53:14.740
So those are well-defined.
00:53:24.170 --> 00:53:26.930
What's the relationship
between L1
00:53:26.930 --> 00:53:30.160
functions and L2 functions?
00:53:30.160 --> 00:53:34.080
Can a function be
L1 and not L2?
00:53:34.080 --> 00:53:35.570
Can it be L2 and not L1?
00:53:39.090 --> 00:53:43.930
Yeah, it can be both,
unfortunately.
00:53:43.930 --> 00:53:45.680
All possibilities exist.
00:53:45.680 --> 00:53:48.870
You can have functions which
are neither L1 nor L2,
00:53:48.870 --> 00:53:52.100
functions that are L1 but not
L2, functions that are L2 but
00:53:52.100 --> 00:53:55.940
not L1, and functions that
are both L1 and L2.
00:53:55.940 --> 00:53:58.250
Those are the truly nice
functions that we
00:53:58.250 --> 00:53:59.850
like to deal with.
00:53:59.850 --> 00:54:04.810
But there's one nice that you
can say, and that follows from
00:54:04.810 --> 00:54:07.450
a simple argument here.
00:54:07.450 --> 00:54:10.370
If u of t is less than or equal
to 1, if the magnitude
00:54:10.370 --> 00:54:12.600
of u of t is less than
or equal to 1--.
00:54:19.390 --> 00:54:22.660
Let's start out by looking at
u of t being greater than or
00:54:22.660 --> 00:54:23.430
equal to 1.
00:54:23.430 --> 00:54:26.950
If u of t is greater than or
equal to 1, then u squared of
00:54:26.950 --> 00:54:29.830
t is even bigger.
00:54:29.830 --> 00:54:32.250
You see the thing that happened
is when u of t
00:54:32.250 --> 00:54:34.980
becomes bigger than
t, u squared of t
00:54:34.980 --> 00:54:36.920
becomes even more bigger.
00:54:36.920 --> 00:54:41.420
When u of t is less than
1, u squared of t is
00:54:41.420 --> 00:54:42.760
less than u of t.
00:54:46.960 --> 00:54:51.330
But if u of t is less than or
equal to 1, it's less than 1.
00:54:51.330 --> 00:54:56.290
So in all cases, for all t, u
of t, magnitude is less than
00:54:56.290 --> 00:55:02.070
or equal to u of t
squared plus 1.
00:55:02.070 --> 00:55:04.090
So that takes into account
both cases.
00:55:04.090 --> 00:55:06.540
It's a bound.
00:55:06.540 --> 00:55:13.500
So if I'm looking at functions
which only exist between over
00:55:13.500 --> 00:55:18.280
some limited time interval, and
I take the integral from
00:55:18.280 --> 00:55:22.480
minus t over 2 to the t over 2
of the magnitude of u of t, I
00:55:22.480 --> 00:55:25.910
get something which is less than
or equal to the integral
00:55:25.910 --> 00:55:31.000
of u squared of t plus the
integral of 1, and the
00:55:31.000 --> 00:55:35.740
integral of 1 over this finite
limit is just t.
00:55:35.740 --> 00:55:44.870
This says that if a function
is L2 and the function only
00:55:44.870 --> 00:55:51.110
exists over a finite interval,
then the function is L1 also.
00:55:51.110 --> 00:55:54.490
So as long as I'm dealing with
Fourier series, as long as I'm
00:55:54.490 --> 00:56:00.270
dealing with finite duration
functions, L2 means L1.
00:56:00.270 --> 00:56:03.550
All of the nice things that
you get with L1 functions
00:56:03.550 --> 00:56:07.760
apply to L2 functions also, and
there are a lot of nice
00:56:07.760 --> 00:56:09.920
things that happen
for L1 functions.
00:56:09.920 --> 00:56:14.340
There are a lot of nice things
that happen for L2 functions.
00:56:14.340 --> 00:56:18.310
You take the union of what
happens for L1 and for L2, and
00:56:18.310 --> 00:56:20.740
that's beautiful.
00:56:20.740 --> 00:56:23.460
You can say anything then.
00:56:23.460 --> 00:56:27.395
Can't calculate anything, of
course, but we all said, we
00:56:27.395 --> 00:56:30.480
leave that to computers.
00:56:30.480 --> 00:56:32.530
Let's go back Fourier
series now, let's go
00:56:32.530 --> 00:56:34.300
back to the real world.
00:56:37.890 --> 00:56:44.450
Any old function we have u of t,
the magnitude of u of t and
00:56:44.450 --> 00:56:49.402
the magnitude of u of t times
either the 2 pi ift, for any
00:56:49.402 --> 00:56:54.490
old f, this thing has magnitude
1, right, a complex
00:56:54.490 --> 00:56:57.060
exponential.
00:56:57.060 --> 00:56:59.620
Real f, real t.
00:56:59.620 --> 00:57:02.400
This just has magnitude 1.
00:57:02.400 --> 00:57:07.360
And therefore, this magnitude
is equal to this magnitude.
00:57:07.360 --> 00:57:12.660
This says that if the function
u of t is L1, then the
00:57:12.660 --> 00:57:21.070
function u of t times either the
2 pi IFT is also L1, which
00:57:21.070 --> 00:57:23.710
says if we can integrate one
we can integrate the other.
00:57:27.060 --> 00:57:31.200
In other words, the integral
of u of t, either the 2 pi
00:57:31.200 --> 00:57:35.190
ift, the magnitude of
the fdt is going to
00:57:35.190 --> 00:57:38.750
be less than infinity.
00:57:38.750 --> 00:57:40.960
Since we're taking the
magnitude, it's either finite
00:57:40.960 --> 00:57:45.520
or it's infinite, and since u
of t in magnitude when we
00:57:45.520 --> 00:57:48.630
integrate it is less than
infinity, this thing is less
00:57:48.630 --> 00:57:50.750
than infinity also.
00:57:50.750 --> 00:57:54.230
Now, this is a complex
number in here.
00:57:54.230 --> 00:57:58.260
So you can break it up into a
real part and an imaginary
00:57:58.260 --> 00:58:04.350
part, and if this whole thing,
if the magnitude is less than
00:58:04.350 --> 00:58:13.250
infinity, then the magnitude
of the real part is finite.
00:58:13.250 --> 00:58:17.430
If you take the real part over
the region where this is
00:58:17.430 --> 00:58:22.110
positive and the region where
it's negative, you still get
00:58:22.110 --> 00:58:23.460
non-negative numbers.
00:58:23.460 --> 00:58:26.160
In other words, if we're
taking the integral of
00:58:26.160 --> 00:58:28.660
something which has
positive values --
00:58:28.660 --> 00:58:33.080
I should have written this out
in more detail, it's not
00:58:33.080 --> 00:58:34.330
enough to--.
00:58:41.790 --> 00:58:44.010
I'm taking the integral
of something which--.
00:58:51.040 --> 00:58:52.290
This is u of t.
00:58:58.740 --> 00:59:00.120
If I can find another color.
00:59:07.340 --> 00:59:10.990
Let me draw magnitude of
u of t on top of this.
00:59:16.510 --> 00:59:22.690
This thing here is magnitude
of u of t.
00:59:22.690 --> 00:59:23.640
What?
00:59:23.640 --> 00:59:25.670
AUDIENCE: [INAUDIBLE].
00:59:25.670 --> 00:59:27.390
PROFESSOR: I'm making it real
for the time being because I'm
00:59:27.390 --> 00:59:28.660
just looking at the real part.
00:59:32.480 --> 00:59:34.840
In other words, what I'm looking
is this quantity here.
00:59:37.610 --> 00:59:41.820
Later you can imagine doing the
same thing out on complex
00:59:41.820 --> 00:59:43.070
numbers, OK?
00:59:45.660 --> 00:59:48.000
So what I'm saying is if I just
look at the real part of
00:59:48.000 --> 00:59:51.390
u of t -- call this real part
of u of t, if you like.
00:59:56.250 --> 01:00:03.390
If I know that this is finite,
this is non-negative, I know
01:00:03.390 --> 01:00:07.670
that the positive part of this
function has a finite
01:00:07.670 --> 01:00:11.300
integral, I know that the
negative part of it has a
01:00:11.300 --> 01:00:13.270
finite integral.
01:00:13.270 --> 01:00:17.170
In other words, the thing which
makes integration messy
01:00:17.170 --> 01:00:21.760
is you sometimes have the
positive part being infinite,
01:00:21.760 --> 01:00:24.860
the negative part being infinite
also, and the two of
01:00:24.860 --> 01:00:27.180
them cancel out somehow to give
you something finite.
01:00:29.680 --> 01:00:32.290
When you're dealing with the
magnitude of u of t, if the
01:00:32.290 --> 01:00:35.960
magnitude of u of t has an
infinite integral, then that
01:00:35.960 --> 01:00:37.880
messy thing can't happen.
01:00:37.880 --> 01:00:42.850
It says that the positive part
has a finite integral, the
01:00:42.850 --> 01:00:45.510
negative part has a finite
integral also.
01:00:45.510 --> 01:00:49.160
If you take the imaginary part,
visualize that out this
01:00:49.160 --> 01:00:53.660
way, the positive part of the
imaginary part has a finite
01:00:53.660 --> 01:00:57.430
integral, the negative part of
the imaginary part has a
01:00:57.430 --> 01:00:59.610
finite integral also.
01:00:59.610 --> 01:01:07.380
It says if the magnitude of u
of t, that integral always
01:01:07.380 --> 01:01:11.920
exists, and if it's finite, then
these positive parts of
01:01:11.920 --> 01:01:15.690
the real, the positive part of
the imaginary part, all of
01:01:15.690 --> 01:01:21.150
those are finite, and it says
the integral itself is finite.
01:01:21.150 --> 01:01:26.130
Which says that this integral
here has to be finite.
01:01:26.130 --> 01:01:29.880
Namely, the positive part of
the real part, the negative
01:01:29.880 --> 01:01:33.460
part of the real part, the
positive part of the imaginary
01:01:33.460 --> 01:01:37.800
part, the negative part of the
imaginary part, all four have
01:01:37.800 --> 01:01:44.840
to be finite, just because this
quantity here is finite.
01:01:44.840 --> 01:01:50.130
Now, if u of 2 is L2 and also
time limited, it's L1, and the
01:01:50.130 --> 01:01:53.020
same conclusion follows.
01:01:53.020 --> 01:01:56.130
So this integral always
exists if it's
01:01:56.130 --> 01:01:57.380
over a finite interval.
01:02:01.780 --> 01:02:05.090
So at this point we're really
ready to go back to this
01:02:05.090 --> 01:02:09.580
theorem about Fourier series
that we stated last time and
01:02:09.580 --> 01:02:13.290
which was a little bit
mysterious at that point.
01:02:13.290 --> 01:02:16.920
In fact, at this point we've
already proven part of it.
01:02:16.920 --> 01:02:20.650
I wanted to prove it because I
wanted you to know that not
01:02:20.650 --> 01:02:23.460
everything in measure
theory is difficult.
01:02:23.460 --> 01:02:27.040
An awful lot of these things,
after you know just a very
01:02:27.040 --> 01:02:31.880
small number of the ideas,
are very, very simple.
01:02:31.880 --> 01:02:34.770
Now, you will think this is not
simple because you haven't
01:02:34.770 --> 01:02:36.895
had time to think about
the ten slides
01:02:36.895 --> 01:02:39.620
that have gone before.
01:02:39.620 --> 01:02:41.920
But if you go back and you look
at them again, if you
01:02:41.920 --> 01:02:44.630
read the notes, you will see
that, in fact, it all is
01:02:44.630 --> 01:02:46.860
pretty simple.
01:02:46.860 --> 01:02:51.760
What this says is if u of t, a
complex function real into c,
01:02:51.760 --> 01:02:58.820
but time limited, suppose it's
an L2 function, then it's also
01:02:58.820 --> 01:03:04.610
L1 over that interval minus
t over 2 to plus t over 2.
01:03:04.610 --> 01:03:10.320
Then for each k and z, then
for each integer k, this
01:03:10.320 --> 01:03:16.930
integral here, this function
here, is now an L1 function,
01:03:16.930 --> 01:03:21.340
and therefore, this integral
exists, is finite.
01:03:21.340 --> 01:03:23.890
You divide by t is
still finite.
01:03:23.890 --> 01:03:28.660
So that Fourier coefficient
has to exist and it has to
01:03:28.660 --> 01:03:30.200
exist as a finite value.
01:03:32.870 --> 01:03:36.220
Now, you look at Reimann
integration, and you look at
01:03:36.220 --> 01:03:39.410
the theorems about Reimann
integration, and if they're
01:03:39.410 --> 01:03:43.280
stated by somebody who was
stating theorems, the
01:03:43.280 --> 01:03:47.150
conditions are monstrous.
01:03:47.150 --> 01:03:49.490
This is not monstrous.
01:03:49.490 --> 01:03:53.870
It says all you need is
measurability and L1, which
01:03:53.870 --> 01:03:56.290
says it doesn't go
up to infinity.
01:03:56.290 --> 01:03:58.490
That's enough to say that every
one of these Fourier
01:03:58.490 --> 01:04:00.830
coefficients has to exist.
01:04:04.830 --> 01:04:07.310
It might be hard to integrate
it, but you
01:04:07.310 --> 01:04:08.960
know it has to exist.
01:04:08.960 --> 01:04:13.180
Now the next thing it says
is that -- this is more
01:04:13.180 --> 01:04:14.880
complicated.
01:04:14.880 --> 01:04:18.870
What we would like to say and
what we tried to say before
01:04:18.870 --> 01:04:22.170
and what you like to say with
the Fourier series is that u
01:04:22.170 --> 01:04:28.270
of t is equal to this, where you
sum from minus infinity to
01:04:28.270 --> 01:04:30.220
plus infinity.
01:04:30.220 --> 01:04:32.420
We saw that we can't say that.
01:04:32.420 --> 01:04:37.670
We saw that we can't it for
functions which have step
01:04:37.670 --> 01:04:41.120
discontinuities, because
whenever you have a step
01:04:41.120 --> 01:04:44.480
discontinuity, the Fourier
series converges to the
01:04:44.480 --> 01:04:47.110
mid-point of that
discontinuity.
01:04:47.110 --> 01:04:51.200
If you were unfortunate enough
to try to make life simple and
01:04:51.200 --> 01:04:58.120
define the function without
defining it at the step
01:04:58.120 --> 01:05:02.000
discontinuity as the mid-point,
then the Fourier
01:05:02.000 --> 01:05:04.510
series would not be
equal to u of t.
01:05:04.510 --> 01:05:07.860
But what this says is that if
you take the difference
01:05:07.860 --> 01:05:14.500
between u of t and a finite
expansion, and then you look
01:05:14.500 --> 01:05:18.550
at the energy in that
difference, it says that the
01:05:18.550 --> 01:05:20.670
energy and the difference
goes to zero.
01:05:23.930 --> 01:05:29.020
Now that's far more important
than having this integral be
01:05:29.020 --> 01:05:34.990
equal to that, because frankly,
we don't care a fig
01:05:34.990 --> 01:05:41.010
for whether this is equal to
this at every t or not.
01:05:41.010 --> 01:05:45.330
What we care about is when we
add more and more terms onto
01:05:45.330 --> 01:05:47.050
this Fourier series.
01:05:47.050 --> 01:05:50.280
I mean in engineering we're
always approximating things.
01:05:50.280 --> 01:05:53.010
We have to approximate things.
01:05:53.010 --> 01:05:56.600
We talk about functions u of
t, but our functions u of t
01:05:56.600 --> 01:06:00.160
are just models of things
anyway, and we want those
01:06:00.160 --> 01:06:04.450
models to really converge to
something, which means that as
01:06:04.450 --> 01:06:07.670
we take more and more terms in
the Fourier series, we get
01:06:07.670 --> 01:06:11.110
something which comes closer to
u of t and it comes closer
01:06:11.110 --> 01:06:12.310
in energy terms.
01:06:12.310 --> 01:06:16.090
Remember when we take this, when
we approximate it in this
01:06:16.090 --> 01:06:19.940
way by a finite Fourier series,
and we then quantize
01:06:19.940 --> 01:06:29.160
coefficients, and then we go
back to the function, we have
01:06:29.160 --> 01:06:33.580
lost all the coefficients and
all of these terms for the
01:06:33.580 --> 01:06:34.840
very high frequencies.
01:06:34.840 --> 01:06:37.090
We've dropped them off.
01:06:37.090 --> 01:06:42.460
What this says is if we take
more and more of them and then
01:06:42.460 --> 01:06:46.490
we quantize and we go back to
a function v of t it says as
01:06:46.490 --> 01:06:50.080
we add more and more Fourier
coefficients and quantize
01:06:50.080 --> 01:06:53.585
carefully, we can come closer
and closer to the function
01:06:53.585 --> 01:06:56.680
that we started with.
01:06:56.680 --> 01:07:01.410
You don't get that by talking
about point to point equality,
01:07:01.410 --> 01:07:05.600
because as soon as we quantize
you lose all of that anyway.
01:07:05.600 --> 01:07:08.640
You don't have a quality
anywhere anymore, and the only
01:07:08.640 --> 01:07:12.560
thing you can hope for is a
small mean square error.
01:07:15.370 --> 01:07:19.230
So, this looks more complicated
than what you
01:07:19.230 --> 01:07:23.360
would like, but what I'm trying
to tell you is that
01:07:23.360 --> 01:07:26.650
this is far more important
than what you would like.
01:07:26.650 --> 01:07:29.190
What you would like is
not important at all.
01:07:29.190 --> 01:07:33.030
I can give you examples of
things where this converges to
01:07:33.030 --> 01:07:37.270
this everywhere, but in fact,
no matter how many terms you
01:07:37.270 --> 01:07:41.390
take, the energy difference
between these two things is
01:07:41.390 --> 01:07:43.480
very large.
01:07:43.480 --> 01:07:45.610
Those are the ugly things.
01:07:45.610 --> 01:07:52.560
That says you never get a good
approximation, even though
01:07:52.560 --> 01:07:56.980
looking at things point-wise
everything looks great.
01:07:56.980 --> 01:07:59.380
But what you're really
interested in is these mean
01:07:59.380 --> 01:08:04.710
square approximations and what
this Fourier series thing says
01:08:04.710 --> 01:08:08.790
is that if you deal with
measurable functions then this
01:08:08.790 --> 01:08:12.960
converges just to that in this
very nice energy sense, and
01:08:12.960 --> 01:08:15.580
energy is what we're
interested in.
01:08:15.580 --> 01:08:16.700
The final part of this --
01:08:16.700 --> 01:08:20.590
I mean I talked about this
a little bit last time --
01:08:20.590 --> 01:08:24.020
sometimes instead of starting
out with a function and
01:08:24.020 --> 01:08:27.010
approximating it with the
Fourier series, you want to
01:08:27.010 --> 01:08:29.140
start out with the coefficients
01:08:29.140 --> 01:08:30.830
and find the function.
01:08:30.830 --> 01:08:34.140
In fact, when we start talking
about modulation, that's
01:08:34.140 --> 01:08:37.270
exactly what we're going to be
doing because we're always
01:08:37.270 --> 01:08:40.250
going to be starting out with
these digital sequences, we're
01:08:40.250 --> 01:08:43.790
going to be finding functions
from the digital sequences.
01:08:43.790 --> 01:08:46.810
This final part of the theorem
says yes, you can
01:08:46.810 --> 01:08:48.380
do that and it works.
01:08:48.380 --> 01:08:53.730
It says that given any set
of coefficients where the
01:08:53.730 --> 01:08:56.550
coefficients have finite energy,
in other words, where
01:08:56.550 --> 01:09:02.070
the sum is less than infinity,
then there's an L2 function, u
01:09:02.070 --> 01:09:08.470
of t, which satisfies the above
in this limiting sense.
01:09:08.470 --> 01:09:10.680
Now this is the hardest thing
of the theorem to prove.
01:09:10.680 --> 01:09:13.840
It looks obvious but it's not.
01:09:13.840 --> 01:09:15.680
But anyway, it's there.
01:09:15.680 --> 01:09:20.790
It says these approximations are
rock solid and you can now
01:09:20.790 --> 01:09:24.820
forget about all of this measure
theoretic stuff, and
01:09:24.820 --> 01:09:28.220
you can just live with the
fact that all of these in
01:09:28.220 --> 01:09:31.340
terms of L2 approximations,
work.
01:09:39.030 --> 01:09:41.510
I mean again, it doesn't look
beautiful at this point
01:09:41.510 --> 01:09:44.380
because you haven't seen
it for long enough.
01:09:44.380 --> 01:09:45.370
It really is beautiful.
01:09:45.370 --> 01:09:47.150
It's beautiful stuff.
01:09:47.150 --> 01:09:52.050
When you think about it long
enough, for 40 years is how
01:09:52.050 --> 01:09:55.140
long I've been thinking about
it, it becomes more and more
01:09:55.140 --> 01:09:56.370
beautiful every year.
01:09:56.370 --> 01:09:59.740
So, if you live long enough
it'll be beautiful.
01:10:04.120 --> 01:10:07.950
Any time you talk about this
type of convergence, we call
01:10:07.950 --> 01:10:15.040
it convergence in the mean, and
then notation we will use
01:10:15.040 --> 01:10:20.180
is l.i.m., which looks like
limit but which really stands
01:10:20.180 --> 01:10:24.140
for limit in the mean.
01:10:24.140 --> 01:10:28.070
We will write that complicated
thing on the last slide, which
01:10:28.070 --> 01:10:31.780
I said is not really that
complicated, but we will write
01:10:31.780 --> 01:10:35.930
this as this.
01:10:35.930 --> 01:10:39.290
In other words, when we write
limit in the mean, we mean
01:10:39.290 --> 01:10:43.250
that the difference between
these two sides in energy
01:10:43.250 --> 01:10:46.360
sense goes to zero
as k gets large.
01:10:46.360 --> 01:10:48.360
That's what this limit means.
01:10:48.360 --> 01:10:51.650
It just means the statement that
we talked about before.
01:10:51.650 --> 01:10:57.420
So the Fourier series, the
theorem really says you can
01:10:57.420 --> 01:11:01.080
find all of the Fourier
coefficients in this way, they
01:11:01.080 --> 01:11:05.610
all exist, they all exist
exactly, they're all finite.
01:11:05.610 --> 01:11:10.530
The function then exists as this
limit in the mean, which
01:11:10.530 --> 01:11:13.250
is the thing that we're
always interested in.
01:11:13.250 --> 01:11:19.760
So, every L2 function defined
over a finite interval, and
01:11:19.760 --> 01:11:23.700
every L1 function defined over
a finite interval, all of
01:11:23.700 --> 01:11:27.420
these have a Fourier series
which satisfies these two
01:11:27.420 --> 01:11:30.930
relationships.
01:11:30.930 --> 01:11:33.000
Now, what we're going to do
with all of this is we're
01:11:33.000 --> 01:11:38.090
going to segment an arbitrary
function, an arbitrary L2
01:11:38.090 --> 01:11:44.460
function over the entire time
range, and we're going to
01:11:44.460 --> 01:11:50.760
segment it into pieces,
all of width t.
01:11:50.760 --> 01:11:54.530
Then we're going to expand each
of those segments into a
01:11:54.530 --> 01:11:55.780
Fourier series.
01:11:58.480 --> 01:12:03.890
That's what people do any time
they compress voice that I
01:12:03.890 --> 01:12:04.960
said several times.
01:12:04.960 --> 01:12:07.060
When you compress voice --
01:12:07.060 --> 01:12:09.810
for some reason or other
everybody does it in 20
01:12:09.810 --> 01:12:11.960
millisecond increments.
01:12:11.960 --> 01:12:15.930
You chop up the voice into 20
millisecond increments.
01:12:15.930 --> 01:12:19.090
You then, at least, conceptually
look at those 20
01:12:19.090 --> 01:12:22.625
millisecond increments, think
of them in terms of the
01:12:22.625 --> 01:12:25.930
Fourier series, you expand them
in the Fourier series.
01:12:25.930 --> 01:12:29.400
So for each increment in time
we get a different Fourier
01:12:29.400 --> 01:12:32.590
series, and we use those Fourier
series to approximate
01:12:32.590 --> 01:12:34.270
the function.
01:12:34.270 --> 01:12:38.990
So, each one of the increments
now is going to be represented
01:12:38.990 --> 01:12:41.170
in this way here.
01:12:41.170 --> 01:12:45.330
Since we're only looking at
the function now over the
01:12:45.330 --> 01:12:51.710
interval around time, mt, we
look at it this way, we
01:12:51.710 --> 01:12:55.540
calculate the coefficients,
again, in terms of this
01:12:55.540 --> 01:13:00.980
rectangular function
spaced off by m.
01:13:00.980 --> 01:13:05.170
This is saying the same thing
that we said before, just
01:13:05.170 --> 01:13:12.530
moving it from minus t over 2
to t over 2 to t minus mt.
01:13:21.990 --> 01:13:24.730
Here's minus t over 2.
01:13:24.730 --> 01:13:27.560
The t over 2.
01:13:27.560 --> 01:13:32.050
Next we're looking at t
over 2 to 3t over 2.
01:13:32.050 --> 01:13:36.280
Next we're looking
at 5t over 2.
01:13:36.280 --> 01:13:39.690
This is m equals zero.
01:13:39.690 --> 01:13:41.990
This is m equals 1.
01:13:41.990 --> 01:13:44.950
This is m equals 2.
01:13:44.950 --> 01:13:49.350
This notation here you should
get used to it, it will be
01:13:49.350 --> 01:13:51.570
confusing for a while.
01:13:51.570 --> 01:13:58.720
All it means is that,
and it works.
01:13:58.720 --> 01:14:01.350
So, in fact, you can take all
these terms, you can break
01:14:01.350 --> 01:14:02.600
them up this way.
01:14:05.120 --> 01:14:08.070
So what you've really done is
you've broken u of t into a
01:14:08.070 --> 01:14:12.180
double sub expansion.
01:14:12.180 --> 01:14:19.150
These exponentials limited in
time, that entire set of
01:14:19.150 --> 01:14:22.320
functions are talking
to each other.
01:14:22.320 --> 01:14:25.020
A function living in this
interval of time and a
01:14:25.020 --> 01:14:27.790
function living in this interval
of time have to be
01:14:27.790 --> 01:14:31.250
orthogonal, because I multiply
this by this and I get
01:14:31.250 --> 01:14:34.570
zero at each time.
01:14:34.570 --> 01:14:39.000
In one interval these
exponentials are orthogonal to
01:14:39.000 --> 01:14:40.940
each other -- we've pointed
that out before.
01:14:40.940 --> 01:14:46.830
So we have a doubly exponential,
we have a double
01:14:46.830 --> 01:14:51.450
sum of orthogonal functions, and
what we're saying is that
01:14:51.450 --> 01:14:56.290
any old L2 function at all
we can break up into
01:14:56.290 --> 01:14:59.940
this kind of sum here.
01:14:59.940 --> 01:15:05.090
This is a very complicated way
of saying what we think of
01:15:05.090 --> 01:15:06.740
physically anyway.
01:15:06.740 --> 01:15:10.020
It says take a big long
function, segment it into
01:15:10.020 --> 01:15:14.390
intervals of length t, break up
each interval of length t
01:15:14.390 --> 01:15:15.680
into a Fourier series.
01:15:15.680 --> 01:15:18.300
That's all that's saying.
01:15:18.300 --> 01:15:20.580
So it's saying what
is obvious.
01:15:20.580 --> 01:15:24.500
We're going to find a number of
such orthogonal expansions
01:15:24.500 --> 01:15:27.200
which work for arbitrary
L2 functions.
01:15:29.920 --> 01:15:32.700
As I said, it's a conceptual
basis for voice compress
01:15:32.700 --> 01:15:34.770
algorithms.
01:15:34.770 --> 01:15:38.480
Even more, next time
we're going to go
01:15:38.480 --> 01:15:40.610
into the Fourier integral.
01:15:40.610 --> 01:15:43.760
You think of the Fourier
integral as being the right
01:15:43.760 --> 01:15:50.290
way to go from time to
frequency, but, in fact, it's
01:15:50.290 --> 01:15:54.670
not really the right way to
go from time to frequency.
01:15:54.670 --> 01:16:00.300
When we think of voice or you
think of the wave form of a
01:16:00.300 --> 01:16:06.310
symphony, for example,
what going on there?
01:16:06.310 --> 01:16:09.780
Over every little interval
of time you hear various
01:16:09.780 --> 01:16:12.770
frequencies, right?
01:16:12.770 --> 01:16:19.260
In fact, if you make t equal
to the timing of the music,
01:16:19.260 --> 01:16:24.220
the idea becomes very, very
clean because at each time
01:16:24.220 --> 01:16:26.090
somebody changes a note.
01:16:26.090 --> 01:16:29.590
So you go from one frequency to
another frequency, so it's
01:16:29.590 --> 01:16:32.290
a rather clean way of
looking at this.
01:16:32.290 --> 01:16:36.860
Our notion of frequency that
we think of intuitively is
01:16:36.860 --> 01:16:42.500
much more closely associated
with the idea of frequencies
01:16:42.500 --> 01:16:45.620
is changing in time then
it is of frequencies
01:16:45.620 --> 01:16:46.880
being constant in time.
01:16:46.880 --> 01:16:52.050
When we look at a Fourier
integral, everything
01:16:52.050 --> 01:16:53.380
is frozen in time.
01:16:53.380 --> 01:16:55.910
As soon as you take the Fourier
integral, you've
01:16:55.910 --> 01:16:59.520
glopped everything from time
minus infinity to time plus
01:16:59.520 --> 01:17:01.610
infinity all together.
01:17:01.610 --> 01:17:04.410
Here when we look at these
expansions, we're looking at
01:17:04.410 --> 01:17:06.880
these frequencies changing.
01:17:06.880 --> 01:17:10.510
There's an unfortunate part
about frequencies changing,
01:17:10.510 --> 01:17:13.410
and that is frequencies,
unfortunately, live over the
01:17:13.410 --> 01:17:14.910
entire time interval.
01:17:14.910 --> 01:17:18.540
These truncated frequencies
work very nicely but they
01:17:18.540 --> 01:17:22.590
don't quite correspond to
non-truncated time intervals,
01:17:22.590 --> 01:17:26.030
but it still does match our
intuition and this is a useful
01:17:26.030 --> 01:17:29.770
way to think about functions
that change in time.
01:17:34.310 --> 01:17:37.840
If you believe what I've just
said, why do people ever worry
01:17:37.840 --> 01:17:39.660
about the Fourier
integral at all?
01:17:42.650 --> 01:17:46.830
Well, you see the problem is
this quantity t here that I'm
01:17:46.830 --> 01:17:52.350
segmenting over is unnatural.
01:17:52.350 --> 01:17:55.990
If I look at voice, there's no t
that you can take which is a
01:17:55.990 --> 01:17:56.720
natural quantity.
01:17:56.720 --> 01:18:00.330
The 20 milliseconds is just
something arbitrary that
01:18:00.330 --> 01:18:03.360
somebody did once
and it worked.
01:18:03.360 --> 01:18:06.780
Too many other engineers in
the field said, well that
01:18:06.780 --> 01:18:09.420
works, I'll pick the same thing
instead of trying to
01:18:09.420 --> 01:18:12.450
think through what a better
number would be.
01:18:12.450 --> 01:18:14.210
So they all use the same t.
01:18:14.210 --> 01:18:16.880
It doesn't correspond to
anything physically.
01:18:16.880 --> 01:18:20.310
So, in fact, people do
try the same thing.
01:18:20.310 --> 01:18:24.100
Let me just say what a Fourier
transform is and we'll talk
01:18:24.100 --> 01:18:27.150
about it most of next time
because that's the next thing
01:18:27.150 --> 01:18:28.260
we have to deal with.
01:18:28.260 --> 01:18:31.510
Something you're all familiar
with are these Fourier
01:18:31.510 --> 01:18:33.900
transform relationships.
01:18:33.900 --> 01:18:36.290
There's the function
of time, there's
01:18:36.290 --> 01:18:38.010
the function of frequency.
01:18:38.010 --> 01:18:43.000
You can go from one to
the other, mapping.
01:18:43.000 --> 01:18:45.640
Well, if you know this
you can find this.
01:18:45.640 --> 01:18:47.930
If you know this, you
can find this.
01:18:47.930 --> 01:18:50.580
It looks a little bit like the
Fourier series, and usually
01:18:50.580 --> 01:18:53.530
when people learn about the
Fourier integral they start
01:18:53.530 --> 01:18:58.050
with a Fourier series and start
letting t become large,
01:18:58.050 --> 01:19:00.700
which doesn't quite work.
01:19:00.700 --> 01:19:04.730
But anyway, that's what
we're going to do.
01:19:04.730 --> 01:19:07.180
If the function is well-behaved,
the first
01:19:07.180 --> 01:19:10.260
integral exists and
the second exists.
01:19:10.260 --> 01:19:12.560
What does well-behaved mean?
01:19:12.560 --> 01:19:14.340
It means what we
usually mean by
01:19:14.340 --> 01:19:18.050
well-behaved, it means it works.
01:19:18.050 --> 01:19:23.640
So again, this theorem here is
another example, or not, as
01:19:23.640 --> 01:19:25.330
the case may be.
01:19:25.330 --> 01:19:28.710
But we'll make this
clearer next time.