1
00:00:16,330 --> 00:00:29,590
This is the prelims course called Analysis three. And this course is about integration.
2
00:00:29,590 --> 00:00:35,320
So let me start by telling you a few things about what's going to be in the course.
3
00:00:35,320 --> 00:00:43,660
So essentially, this course is about making rigorous the notion of an integral.
4
00:00:43,660 --> 00:01:02,520
So how can we make the notion of an integral, rigorous cost control?
5
00:01:02,520 --> 00:01:03,410
And so basically,
6
00:01:03,410 --> 00:01:14,200
what it was I'm going to be doing is showing you how to make rigorous sense of lots of facts that I'm sure you've all known since school.
7
00:01:14,200 --> 00:01:23,070
So we will make rigorous sense.
8
00:01:23,070 --> 00:01:50,460
Of facts such as. Well, things like that, the integral of X squared from zero to one is one third and even what that means and then things,
9
00:01:50,460 --> 00:01:58,620
I'm sure you all aware of that the basic fact that integration and differentiation opposite operations,
10
00:01:58,620 --> 00:02:12,000
so integration and differentiation are opposites.
11
00:02:12,000 --> 00:02:19,760
So we'll formulate that really carefully and prove a couple of versions of it.
12
00:02:19,760 --> 00:02:27,380
And then we'll also look at some basic functions, which are best defined using integration.
13
00:02:27,380 --> 00:02:41,370
So we'll look at the exponential and logarithm functions.
14
00:02:41,370 --> 00:02:44,850
And then we'll finish by looking at some things that you've not only known since secondary school,
15
00:02:44,850 --> 00:02:56,650
but since primary school, such as the fact that the circumference of a circle is two PI.
16
00:02:56,650 --> 00:03:16,860
So the circumference of the circle of Radius one is two PI and that the area of a circle of Radius one is PI.
17
00:03:16,860 --> 00:03:23,340
So actually today to even define what the circumference or the area of a shape is,
18
00:03:23,340 --> 00:03:31,530
unless it's a sort of rectangle or something like that, you really need to know what an integral is and actually what's PI?
19
00:03:31,530 --> 00:03:37,230
So we'll also be carefully defining PI and showing that it's equal to these two things.
20
00:03:37,230 --> 00:03:47,260
When those two things are carefully defined. So these are some of the things that are going to be in the course of time.
21
00:03:47,260 --> 00:03:56,990
So another comment I want to make. It's a bit of an apology, really.
22
00:03:56,990 --> 00:04:03,980
I'm apologising not for myself yet, but for the subjects of mathematics, really,
23
00:04:03,980 --> 00:04:09,470
so the integral that we're going to talk about in this course is not the right integral.
24
00:04:09,470 --> 00:04:27,020
So we'll be talking about. What's called the Raymond Integral?
25
00:04:27,020 --> 00:04:32,000
And it has the advantage that it's quite intuitive.
26
00:04:32,000 --> 00:05:05,670
It's relatively intuitive. And also relatively easy to define, so relatively, but it has some shortcomings, which we'll also be discussing.
27
00:05:05,670 --> 00:05:16,570
So it has various shortcomings, not the least of which is that there are some functions that you'd like to be able to integrate, but you can't.
28
00:05:16,570 --> 00:05:27,130
And as a result of that, I think it's fair to say that professionals would always use what's called the LeBec integral.
29
00:05:27,130 --> 00:05:41,910
So professionals, whoever they are, tend to use the LeBec in school.
30
00:05:41,910 --> 00:05:53,370
But then the package goes quite a bit more difficult to define. So what why are we spending eight hours talking about this second rate integral?
31
00:05:53,370 --> 00:05:59,250
Well, I think I've already mentioned it's relatively intuitive and easy to define,
32
00:05:59,250 --> 00:06:03,540
but thankfully whenever this remittance bill exists, it's equal to the baggage.
33
00:06:03,540 --> 00:06:11,550
So we are actually talking about a subset of the the kind of full theory that professional mathematicians would use.
34
00:06:11,550 --> 00:06:19,690
So this is a little bit like having to learn to crawl before you can learn to walk.
35
00:06:19,690 --> 00:06:24,930
Now, an apology that's more of a personal apology, but it's not really an apology, just a comment.
36
00:06:24,930 --> 00:06:47,850
I'm actually not going to talk about the remaining screw. I'm actually going to talk about something called the taboo in school.
37
00:06:47,850 --> 00:06:52,300
And the reason I'm going to do that is that I find this the the best way to develop the theory.
38
00:06:52,300 --> 00:06:55,950
I find it just the most intuitive way to talk about the theory.
39
00:06:55,950 --> 00:07:03,470
And it's then a theorem that will prove later in the course that the top goal is the same as the remaining school.
40
00:07:03,470 --> 00:07:22,370
So this turns out to be the same. As the minutes grow, but that's a theorem that has to be proven.
41
00:07:22,370 --> 00:07:30,560
So one consequence of this feature that I'm going to talk about something that's not exactly the same as three minutes grill to begin with,
42
00:07:30,560 --> 00:07:37,550
I would urge you to use the utmost caution if you look at any books about this subject early in the course,
43
00:07:37,550 --> 00:07:44,020
so you don't really need to because there are four lecture notes for the course on the internet.
44
00:07:44,020 --> 00:07:50,170
But different books take subtly different approaches, and it will get very confusing,
45
00:07:50,170 --> 00:07:55,640
at least to begin with if you try and compare what I'm talking about with some books.
46
00:07:55,640 --> 00:08:05,660
What would be good is at the end of the course to have a look at a couple of books and see different ways of developing the theory that,
47
00:08:05,660 --> 00:08:14,450
OK, so I think that's what I would say by way of introduction. So let's make a start on some actual mathematics.
48
00:08:14,450 --> 00:08:33,620
So the first chapter is about step functions and the integral.
49
00:08:33,620 --> 00:08:38,120
So as I'm sure you all aware, the integral, whatever it is of a function,
50
00:08:38,120 --> 00:08:46,910
is supposed to be kind of measuring the area under the graph of that function, whatever that means.
51
00:08:46,910 --> 00:08:55,520
And so what I'm going to do to begin with is look at some extremely basic functions and write down what they're integral surely would have
52
00:08:55,520 --> 00:09:10,200
to be for any vaguely sensible definition of integral so that these functions that I'm going to look at are called the step functions.
53
00:09:10,200 --> 00:09:27,270
So here is the definition, a function fi from a b to the rails, so I'll only be talking about real value functions in this course,
54
00:09:27,270 --> 00:09:41,580
but one could talk about complex value functions without too much more difficulty. So this is called a step function.
55
00:09:41,580 --> 00:10:03,160
If there are some intermediate points between A and B. So a sequence of points, the first of which is and the last of which is B.
56
00:10:03,160 --> 00:10:26,440
I'm such that fight is constant on the open interval between X Y and Z plus one.
57
00:10:26,440 --> 00:10:33,220
For between zero and minus one. So that's all it is.
58
00:10:33,220 --> 00:10:47,310
I'll give a simple example at the moment, but just a quick note I haven't bothered to say anything about the value of fight at the Points X exi.
59
00:10:47,310 --> 00:10:54,880
So we don't care. What value?
60
00:10:54,880 --> 00:11:05,360
Fire takes at the south points outside.
61
00:11:05,360 --> 00:11:09,260
So just to make absolutely clear that we know what we're talking about.
62
00:11:09,260 --> 00:11:22,520
Let me give an example if I take, let's say, eight equals zero and vehicles two.
63
00:11:22,520 --> 00:11:36,300
So the function fly from zero up to two to the rear is defined as follows.
64
00:11:36,300 --> 00:11:56,240
So let's make it 10, it X equals zero. I don't know, three, if X lies between zero and one minus seven at X equals zero.
65
00:11:56,240 --> 00:12:01,790
Sorry, X equals one. I'm one.
66
00:12:01,790 --> 00:12:08,300
If Pax lives between one and two and then minus six, if X equals two.
67
00:12:08,300 --> 00:12:21,230
So that's a step function and we could try and draw the graph of that just to really belabour the point.
68
00:12:21,230 --> 00:12:27,410
So here is supposed to be some axes that zero this one, that's two.
69
00:12:27,410 --> 00:12:36,730
And then this function is going to look a bit like this, so it will be 10 zero.
70
00:12:36,730 --> 00:12:44,050
Three, on this open interval, there are minus seven down there.
71
00:12:44,050 --> 00:12:48,700
One here. And then minus six down here.
72
00:12:48,700 --> 00:12:59,130
So that's the graph of that step function, so that's what they all look like.
73
00:12:59,130 --> 00:13:08,460
We call a sequence of points that splits the interval AB into finally many parts that's called a partition.
74
00:13:08,460 --> 00:13:29,050
So a sequence of points. A equals x nought, that's three x one B is called a partition.
75
00:13:29,050 --> 00:13:40,970
Kelly P of the INS for Abby.
76
00:13:40,970 --> 00:13:57,430
And so we say that the step function is adapted to this particular politician.
77
00:13:57,430 --> 00:14:00,640
OK, so hopefully everybody is pretty comfortable with this definition,
78
00:14:00,640 --> 00:14:05,710
I've gone over it quite slowly because it's going to be absolutely fundamental in the course.
79
00:14:05,710 --> 00:14:17,790
We'll just be talking about step functions over and over again because they're what's used in defining the integral.
80
00:14:17,790 --> 00:14:25,020
And just before stating a simple lemma, I want to wait one more definition, which is a very natural one.
81
00:14:25,020 --> 00:14:29,880
And this is the notion of a politician being a refinement of another politician.
82
00:14:29,880 --> 00:14:35,370
And that's simply that you if you start with a politician such as zero one two here,
83
00:14:35,370 --> 00:14:42,120
then a refinement of it comes by just putting a few more points exi and keeping the original ones.
84
00:14:42,120 --> 00:14:56,490
So definition a politician primed.
85
00:14:56,490 --> 00:15:18,240
So that's going to consist of points Exide primed up to X and prime primed, and that's a refinement of P.
86
00:15:18,240 --> 00:15:28,800
If every X prise is an X, so if every X I primed is an X J.
87
00:15:28,800 --> 00:15:37,510
So that's the notion of refinement. It says clock on console is wrong.
88
00:15:37,510 --> 00:15:43,540
Is there a clock that's actually right anywhere? Does anyone have an opinion on what the time is right now?
89
00:15:43,540 --> 00:15:51,480
The 10 19. OK, so I'm going to make the assumption that this clock is wrong, but sort of consistently wrong.
90
00:15:51,480 --> 00:16:01,430
That's 17 minutes fast. So I'm going to stop the lecture at about 11 12, according to this clock.
91
00:16:01,430 --> 00:16:15,230
OK, let's record. A simple lemma about politicians.
92
00:16:15,230 --> 00:16:34,630
But his records and that. OK, so this is just very simple, basic facts, really, just to check that we've understood what the definitions are.
93
00:16:34,630 --> 00:17:04,700
So suppose that fi is a step function adapted to some politician p then if I take a refinement of p p primes.
94
00:17:04,700 --> 00:17:28,150
Then fire is also a step function adapted to be primed.
95
00:17:28,150 --> 00:17:44,620
Second, if I've got two different politicians, then there's a common refinement of both of them.
96
00:17:44,620 --> 00:17:55,560
So if I've got two politicians of the IDs for Abby, then there's another politician that refines them both.
97
00:17:55,560 --> 00:18:06,030
Common refinement. Pay.
98
00:18:06,030 --> 00:18:16,100
And then finally, just some. Closure properties of the space of step functions.
99
00:18:16,100 --> 00:18:24,270
If five, one and five two step functions.
100
00:18:24,270 --> 00:18:53,980
Then so are. Five, one plus five to scale and multiples, and then also things like the maximum nine five one five two.
101
00:18:53,980 --> 00:19:02,730
So this is this basically nothing to prove here? I'll make a remark about point three.
102
00:19:02,730 --> 00:19:13,920
And I'll talk about the other ones. So I think point one, it's basically obvious it's the case of recording what the definitions are.
103
00:19:13,920 --> 00:19:21,540
Point two is also basically obvious. You just take the points defining the position p one and the points defining the partition
104
00:19:21,540 --> 00:19:27,300
p two and throw them all in together to get your new set of partitioning points.
105
00:19:27,300 --> 00:19:35,520
And then three does deserve a little bit of a remark because these step functions may well be adapted to different politicians.
106
00:19:35,520 --> 00:19:46,560
So I'm going to just say that one and two are pretty obvious. And then three?
107
00:19:46,560 --> 00:20:01,170
Well, I should at least say which politicians are going on here, so I suppose that PHI is adapted to PI.
108
00:20:01,170 --> 00:20:05,100
I equals one two.
109
00:20:05,100 --> 00:20:15,090
Well, then by passing two a common refinement of P1 and P2, I can assume that five one and five two are relative to the same partition.
110
00:20:15,090 --> 00:20:24,610
So by passing to a common refinement. P.
111
00:20:24,610 --> 00:20:29,190
I can assume.
112
00:20:29,190 --> 00:20:40,740
That P1 equals P2 equals pain, and then point three is obvious because it's basically the fact that if you've got two constant functions,
113
00:20:40,740 --> 00:20:47,450
then there some is constantly scale multiple is constant in the max and the constant.
114
00:20:47,450 --> 00:21:10,490
So then this is obvious. Just looking at each of the sub intervals in the partition.
115
00:21:10,490 --> 00:21:17,510
OK, so there's a basic lemma, but the point of which is to say that this is a sort of a reasonable class of functions,
116
00:21:17,510 --> 00:21:43,480
at the least, it is closed on to some of the basic operations on functions. So the next lemma or just before taking the next lemma make a definition.
117
00:21:43,480 --> 00:21:59,110
If I've got a Set X on the rail line, then it's indicator function is just the function that takes the value one on X and zero elsewhere.
118
00:21:59,110 --> 00:22:19,140
It's indicator function. One Sub X is the function of taking values.
119
00:22:19,140 --> 00:22:25,200
Value one for X and X and zero.
120
00:22:25,200 --> 00:22:31,990
If X is not an X. That's quite a standard definition.
121
00:22:31,990 --> 00:22:39,700
And then the Lima number one point two is just the statement that the step functions
122
00:22:39,700 --> 00:22:54,310
are the same thing as the vector space spanned by indicator functions of intervals.
123
00:22:54,310 --> 00:23:16,220
So the space of step functions on a B equals the space of linear combinations, finite linear combinations, of course.
124
00:23:16,220 --> 00:23:32,100
Of indicator functions of intervals.
125
00:23:32,100 --> 00:23:43,710
So One Direction, I think, is really obvious, and that that's the fact that if you've got the indicator function of an interval is a step function,
126
00:23:43,710 --> 00:23:51,440
I think that is obvious from the definition of step function.
127
00:23:51,440 --> 00:24:11,860
So the indicator function of an interval. This is a step function, and I think that's obvious.
128
00:24:11,860 --> 00:24:21,730
And therefore, by the last lemma, a linear combination of indicator functions of intervals is also a step function.
129
00:24:21,730 --> 00:24:43,850
Hence, by Lemma 1.1, the third part, so is any linear combination.
130
00:24:43,850 --> 00:24:46,550
So that's one direction, and then for the other direction,
131
00:24:46,550 --> 00:24:54,470
I need to know that a step function is a linear combination of indicator functions of intervals.
132
00:24:54,470 --> 00:25:07,850
So conversely, a step function is adapted to a particular politician.
133
00:25:07,850 --> 00:25:28,210
So P. Well, it's by definition it is constant on the open intervals, excite up to excite plus one.
134
00:25:28,210 --> 00:25:48,560
And so it's a linear combination. Of while the indicator functions of the open intervals.
135
00:25:48,560 --> 00:26:02,390
And. The trivial intervals, which are just the points, exi.
136
00:26:02,390 --> 00:26:08,360
So notice that I was quite careful not to say whether my intervals were open or closed or half open or half closed,
137
00:26:08,360 --> 00:26:15,920
and so it's important here that I've allowed both open intervals and these closed intervals that consists of just one point.
138
00:26:15,920 --> 00:26:24,450
So let's just make that completely clear. Open interval.
139
00:26:24,450 --> 00:26:36,180
And this is a rather trivial closed interval.
140
00:26:36,180 --> 00:26:41,950
So that's a proof of dilemma.
141
00:26:41,950 --> 00:26:49,840
So occasionally, this space of step functions, which I've not described in two different ways, just straight the basic definition.
142
00:26:49,840 --> 00:26:58,700
And as the linear span of indicator functions of intervals sometimes but not that often given the name.
143
00:26:58,700 --> 00:27:18,420
So sometimes we write our sub step of AB for the facts space of step functions.
144
00:27:18,420 --> 00:27:29,150
On Abe. So let me test my clock hypothesis is the time now 10 30.
145
00:27:29,150 --> 00:27:43,850
Good. So general comments about this course is one of these bits of maths where.
146
00:27:43,850 --> 00:27:50,450
Once one has to go through a lot of very slightly tedious, really and quite simple lemon.
147
00:27:50,450 --> 00:27:56,600
But by the time you've gone through them all, there are so many of them that what you've ended up with is actually not that simple.
148
00:27:56,600 --> 00:28:08,040
So it's one of these things where locally very little happens, but globally. After a few lectures will have ended up with quite a nice theory.
149
00:28:08,040 --> 00:28:12,360
So that section was all about just what a step function is.
150
00:28:12,360 --> 00:28:19,500
And now I want to tell you what the integral of a step function should be, but I'm not going to phrase it like that to begin with.
151
00:28:19,500 --> 00:28:32,200
And so this section is called I of a step function.
152
00:28:32,200 --> 00:28:39,100
So just to make it clear that I'm not using some weird grammar, I mean, I as a function so associated to any step function,
153
00:28:39,100 --> 00:28:45,190
I'm going to assign a quantity called I, which I secretly know is going to be the integral,
154
00:28:45,190 --> 00:28:58,990
but I'm not going to allow myself to call it that just yet. So let me define what I'm talking about.
155
00:28:58,990 --> 00:29:08,920
So suppose I have a step function, so let five be a step function.
156
00:29:08,920 --> 00:29:29,880
Adapt it to. A politician, P.
157
00:29:29,880 --> 00:29:39,430
Between the on on a up to B and let us suppose so, by definition, it's constant between X, Y and Z.
158
00:29:39,430 --> 00:29:53,000
Plus one. I see in the notes I've used x minus one up to excite, so let me do that, that's maybe a bit tidier.
159
00:29:53,000 --> 00:30:00,050
So fi is constant on X.
160
00:30:00,050 --> 00:30:11,400
I'm minus one up to X I four. I equals one up and I let the value that it takes that b c i.
161
00:30:11,400 --> 00:30:21,970
So I suppose fi takes the value. See, I on that interval.
162
00:30:21,970 --> 00:30:37,820
Then we define. I off.I seems to be the sum from I was once and.
163
00:30:37,820 --> 00:30:46,080
Of CII times x minus X on minus one.
164
00:30:46,080 --> 00:31:08,550
So let me compute what is, for this example, function here.
165
00:31:08,550 --> 00:31:20,190
So here there are is to I guess let's just carefully write it down x nought equals not x one equals one x to equals two.
166
00:31:20,190 --> 00:31:34,330
And so I should b c one x one minus x nought plus c to x to minus x one.
167
00:31:34,330 --> 00:31:40,750
So what see, one, that's three. So that's three times one what?
168
00:31:40,750 --> 00:31:48,190
See, two. That's one one times one, which is equal to four.
169
00:31:48,190 --> 00:31:52,330
So if this function is for the what's the point of this definition?
170
00:31:52,330 --> 00:32:01,840
Well, hopefully we would agree that four is the only reasonable value of the integral of this function that I've drawn here.
171
00:32:01,840 --> 00:32:08,420
If somebody said to you, what is the integral of this function, the area under this function?
172
00:32:08,420 --> 00:32:24,590
I haven't defined it yet, but I want you to tell me as a sensible guess as to what it should be, I think the only possible thing you could say is for.
173
00:32:24,590 --> 00:32:47,670
So let me make some remarks. So the first remark is that I have fires what the integral of that function should be.
174
00:32:47,670 --> 00:32:53,520
Where should means, according to any reasonable intuition that you might have?
175
00:32:53,520 --> 00:33:11,540
But we haven't defined the interior yet. And the other remark I want to make, which is, I think again, something that all of you regard is intuitive,
176
00:33:11,540 --> 00:33:24,890
but it's perhaps a little bit less obvious is that the values of fight at the point I make absolutely no contribution to what this includes.
177
00:33:24,890 --> 00:33:44,560
So note that EFI is totally insensitive to the value of PHI at the point at the politician points.
178
00:33:44,560 --> 00:33:49,030
Exile, and I think that's something that is reasonably intuitive, I mean,
179
00:33:49,030 --> 00:34:04,370
they occupy areas zero those points and so they shouldn't make any contribution to the integral if the definition of integral is a sensible one.
180
00:34:04,370 --> 00:34:20,190
And there is a small subtlety to this definition.
181
00:34:20,190 --> 00:34:28,320
And the subtlety is that I've written ify as if this is a quantity that can be defined for any step function.
182
00:34:28,320 --> 00:34:55,060
But the definition uses the underlying politician pay. So the definition of EFI seems to.
183
00:34:55,060 --> 00:35:14,450
Depend on the underlying politician, P, however, that is an illusion.
184
00:35:14,450 --> 00:35:31,080
So why is that an illusion? Well, if I got really sort of overly precise and wrote, I have five comma p instead.
185
00:35:31,080 --> 00:35:47,180
So if we right eye of fi semicolon p for the quantity written above.
186
00:35:47,180 --> 00:35:56,350
Well, then the first thing one can see is that if I pass to a sub to a refinement of pay, it won't change the value.
187
00:35:56,350 --> 00:36:06,940
So I of five primed is equal to I of five P for any refinement.
188
00:36:06,940 --> 00:36:18,580
Of P. So why is that? Well, again, an example, I think it's pretty obvious if I passed a refinement, so I put the point three halves in here.
189
00:36:18,580 --> 00:36:27,530
I think it's reasonably obvious that that's not going to change the value that will still be for so now be something.
190
00:36:27,530 --> 00:36:40,460
Three. Plus, a half plus a half, which is four, so you could, if you want, write down a really careful proof of that, but I think that's pretty clear.
191
00:36:40,460 --> 00:36:45,200
And then if you've got two politicians, P1 and P2, you can pass to a common refinement if both of them.
192
00:36:45,200 --> 00:37:09,900
And so the values of those have got to be equal as well. So now if P1, P2 or any two politicians relative to which fires adapted.
193
00:37:09,900 --> 00:37:20,470
Take a common refinements of both of them. So let P be a common refinement of both of them.
194
00:37:20,470 --> 00:37:36,790
Common refinement of both, and then well, I have fi P. One is I have five, he is I have five two.
195
00:37:36,790 --> 00:37:46,450
And so it doesn't actually matter which politician I take and therefore the definition of I is well defined.
196
00:37:46,450 --> 00:38:01,100
So that's the choice of politician. Choice of politician is immaterial.
197
00:38:01,100 --> 00:38:31,930
And therefore, I have it is indeed well-defined. OK, one more little comment on this section, which is a in fact.
198
00:38:31,930 --> 00:38:39,330
And this is the fact that I is a linear functional on the space of step function, so I,
199
00:38:39,330 --> 00:38:55,230
from all step of AB to the rails is a linear functional, which is just a fancy way of saying.
200
00:38:55,230 --> 00:39:09,030
What basically that is. I mean, it's a linear map between vector spaces, so I of Lambda Phi one plus Ne Phi two is equal to lambda.
201
00:39:09,030 --> 00:39:16,250
I have five one plus mu higher Phi two.
202
00:39:16,250 --> 00:39:25,400
And the proof. Well, it's basically the same as the previous one point one if you pass to a common refinement.
203
00:39:25,400 --> 00:39:32,270
So if five everyone's adapted to partition one and five to two to pursue a common refinement of both.
204
00:39:32,270 --> 00:39:48,740
So they're adapted to the same partition and then it's just obvious so by passing to a common refinement.
205
00:39:48,740 --> 00:40:11,990
We may assume. That's five one and five two are adapted to the same politician pay.
206
00:40:11,990 --> 00:40:35,080
And then it's, I think, reasonably obvious. OK, so that is the definition of I, and now we can finally define what the integral is.
207
00:40:35,080 --> 00:41:02,710
So this will be another new section. So definition of the integral.
208
00:41:02,710 --> 00:41:09,350
So the way not now I'm going to look at arbitrary functions.
209
00:41:09,350 --> 00:41:20,290
And I'm going to try and assign some meaning to what they're integral over the interval AB should be.
210
00:41:20,290 --> 00:41:27,280
So I think this is definitely a point at which we're doing something different to what you've seen, probably at school,
211
00:41:27,280 --> 00:41:35,500
when is used to being given a function f as a kind of formula, maybe a polynomial or as a trigonometric function or some rational function.
212
00:41:35,500 --> 00:41:43,540
And you integrate it using some rules. But now we're just taking an abstractly defined function that may very well not be given by any useful formula.
213
00:41:43,540 --> 00:41:49,230
And we're going to try and say what's meant by its integral.
214
00:41:49,230 --> 00:41:55,830
And the way we would do that is to sandwich f between step function, so here's a picture of what we're going to try and do.
215
00:41:55,830 --> 00:42:03,840
So this is maybe if it doesn't have to be continuous by any means, but I've drawn a continuous function.
216
00:42:03,840 --> 00:42:12,040
And what we're going to try and do is to put step functions above it.
217
00:42:12,040 --> 00:42:30,470
Like that? So a step function that sits above Earth is going to be called a major and by a major and.
218
00:42:30,470 --> 00:42:36,760
Five plus five plus.
219
00:42:36,760 --> 00:42:44,640
For all I mean, is a step function that sits above f point wise.
220
00:42:44,640 --> 00:43:02,880
So we mean a step function. With F of X less than a week, four to five plus of X for all X.
221
00:43:02,880 --> 00:43:22,080
And then, of course, a monument is the same concept, but below. So this and monument would be something like this to not be the same.
222
00:43:22,080 --> 00:43:43,360
So a moment five minus four f, we mean the same thing, a step function with No.
223
00:43:43,360 --> 00:43:49,470
Five minus of X is less than or equal to F of X for all X.
224
00:43:49,470 --> 00:44:09,860
And so I think the picture is maybe self-explanatory. So now we say, let's say the f is integral, if I can find monuments and measurements for it,
225
00:44:09,860 --> 00:44:18,830
whose eye values are arbitrarily close together and the mathematically formalised way of writing that is as follows.
226
00:44:18,830 --> 00:44:28,810
So we say. I say the EFF is integral for.
227
00:44:28,810 --> 00:44:42,640
So I guess I'll say women insurable probably should be taboo win Super Bowl.
228
00:44:42,640 --> 00:45:03,700
If. The SOP over maintenance of I have five minus is equal to the over measurements of I have five plus.
229
00:45:03,700 --> 00:45:15,610
So where just to be clear, the SAP is over minus.
230
00:45:15,610 --> 00:45:28,480
Five minus four F and the inch is over measurements.
231
00:45:28,480 --> 00:45:39,780
I plus. And because this is just such an important definition or in the course.
232
00:45:39,780 --> 00:45:43,620
Let me just be completely clear about what Suffern doing here.
233
00:45:43,620 --> 00:45:46,770
What this means is that for any upsilon greater than zero,
234
00:45:46,770 --> 00:46:02,140
I can find a measurement five plus and a monument fund minus such that the difference of their values is that most asylum equivalently.
235
00:46:02,140 --> 00:46:14,120
So for any asylum crisis in Syria, there exists. So whenever I write this, I will mean monuments and major,
236
00:46:14,120 --> 00:46:29,630
and so there's a monument five minus and a measurement five plus with I have five plus minus, I find minus.
237
00:46:29,630 --> 00:46:45,070
Less than a week two asylum. OK, so that's what it means to be integral.
238
00:46:45,070 --> 00:46:55,830
What is the integral? So the integral is defined to be the common value of these two quantities.
239
00:46:55,830 --> 00:47:14,060
So the integral now we write that it's integral from A to B of F is defined to be.
240
00:47:14,060 --> 00:47:25,730
The common value. Star.
241
00:47:25,730 --> 00:47:34,040
So I think it's it's an intuitively appealing definition. We've defined somehow what the area is if these very basic stump functions called
242
00:47:34,040 --> 00:47:38,330
step functions and to try and define the integral of an arbitrary function.
243
00:47:38,330 --> 00:47:58,620
I'm going to try an approximated above and below by step functions in sandwich it between them, hopefully arbitrarily closely.
244
00:47:58,620 --> 00:48:12,070
OK, so a few little comments about this, keep that that.
245
00:48:12,070 --> 00:48:21,930
So the first comment is, in this case, we're only ever going to be working with bounded functions on A and B.
246
00:48:21,930 --> 00:48:23,320
Well,
247
00:48:23,320 --> 00:48:29,260
except right at the end where we'll talk about something called an improper integral but improper and schools are really just the kind of language,
248
00:48:29,260 --> 00:48:40,930
rather than something that's mathematically rigorous. So we'll always be dealing.
249
00:48:40,930 --> 00:48:55,650
We find it functions. F.
250
00:48:55,650 --> 00:49:06,360
So there's some universal uniform bound on a up to be.
251
00:49:06,360 --> 00:49:18,330
And. And what that means is that there is at least one major and one minor and four F.
252
00:49:18,330 --> 00:49:29,380
So in this case, if has a major hint.
253
00:49:29,380 --> 00:49:46,900
So, namely, the function that's just air everywhere. And it has a minor and.
254
00:49:46,900 --> 00:50:10,160
So that's five minus equals minus M everywhere. So that means at least up in the air, at least exist, so for banded functions.
255
00:50:10,160 --> 00:50:25,530
The infamous up exist.
256
00:50:25,530 --> 00:50:34,820
The other thing to say is that. If I have a minor inch in measurement, so if I've got five minus and five plus.
257
00:50:34,820 --> 00:50:45,990
Then the value of the monument is bounded above by the value of the measurement.
258
00:50:45,990 --> 00:51:09,060
So if five minus and five plus are minor into measurement. Then I of five minus is less than or equal to I have five plus.
259
00:51:09,060 --> 00:51:29,530
And the reason for this is that because of the definitions of minorities in measurement. Five minus is bounded above by five plus point was.
260
00:51:29,530 --> 00:51:36,760
And so if you pass to a common politician relative to which both of these are adopted,
261
00:51:36,760 --> 00:51:43,360
it's then just obvious that their values satisfy the same inequality.
262
00:51:43,360 --> 00:52:06,400
So passing to a politician, a p relative to which fi minus and five plus are both adopted.
263
00:52:06,400 --> 00:52:25,010
It's clear this this inequality. So the fact that there are values I have five minutes is less than a week to, I have five plus is clear.
264
00:52:25,010 --> 00:52:30,660
And so what this means. Because that's true for all minorities, five miners and all major,
265
00:52:30,660 --> 00:52:41,320
it's five plus then the left hand side here is always at most the right hand side here.
266
00:52:41,320 --> 00:53:00,720
So it's up over five minus five minus is at most over five plus I have five plus.
267
00:53:00,720 --> 00:53:18,310
And I guess that also means that if the function is insurable. Which is not always the case, but if essential, could.
268
00:53:18,310 --> 00:53:33,540
Then I of five minus is less than or equal to the integral is less than eight because I have five plus for any five minus and five plus.
269
00:53:33,540 --> 00:53:40,950
OK, so I've just written down a load of more or less obvious facts. It is no rental arithmetic.
270
00:53:40,950 --> 00:53:45,000
Is it in fact? Ten fifty seven. Oh,
271
00:53:45,000 --> 00:53:49,440
that's a shame because I was going to I was going to give an example of a
272
00:53:49,440 --> 00:53:57,870
function that's integral that isn't just a step function that I don't have time.
273
00:53:57,870 --> 00:54:05,580
So I think we'll leave it there for today, and next time, I will at least give an example of a function that's insurable.
274
00:54:05,580 --> 00:54:28,255
Also, the function that is not insurable. And then we'll start proving a few more substantial theorems.