OK, so
Welcome to our first little section on measure theory, which is from
Chapter Three in this module.
And we're going to start
with Section 3.1 on the extended real line,
which is your big chance to
learn how to add and multiply by infinity.
All the things that you were told, never do this,
or you'll be marked down because it's
illegal and incorrect and you mustn't do it.
You're going to have the pleasure of being
allowed to work with infinity correctly.
Even better,
the best thing that comes out of all this is that you're going
to discover that n tends to infinity as n tends to infinity,
which is a truly remarkable fact, that you would never have guessed.
Now we have limited time in this module.
We won't be able to go through all the details,
but I did do some more details when I taught this module to
the third years, well when I taught measure and integration
to the third years.
If you look at my ...
this is a short link to a much longer link which is on my blog,
which then links to various places where
you can get the audio and ... no video unfortunately
yet, but you can get audio podcasts from my lectures from
2006 and 2007 on measure and integration
and all the annotated slides and everything.
So there's plenty of information there,
much more than you need to know.
You can't possibly go away
and learn the whole of a 15 credit,
a fairly difficult 15 credit level three module in
the middle of a 20 credit level four module. OK.
But what there is there is
some additional background material that you might find helpful,
if there's something here where you'd like
a few additional explanations. And the warning on there
that I did used to call sigma-algebras sigma-fields.
They're all ... there's lots of different names for the same thing
but I found that sigma-algebra is more common in the literature,
so we'll call them sigma-algebras this year,
like I did in Functional Analysis last year.
So we're on to Section 3.1 on the extended real line.
What is the extended real line?
It's the closed interval from
minus infinity up to infinity inclusive,
which means, roughly speaking,
you take the real numbers and you need to
find two more things to throw in.
Now I don't care what those things really are.
We're going to call them plus infinity and
minus infinity, or ...I'll often ...
if I say plus infinity it's the same as infinity
in this module and then there's
minus infinity. And that's just notational,
which just means two more things that weren't in
the real line and which we are going
to think of as living at either end,
but it doesn't actually matter what those
two things are, as long as you've got a ...
as long as you've managed to find
a slightly bigger set which has got the real numbers in,
and two more points.
And now we're going to make
some just formal definitions
so that we can define a structure on it.
Just like if you've got any set, you can,
you can find some suitable algebraic structure
or anything that you can put on it,
that's what we're going to do now.
Real line, two more points,
and now we will define some structures on it as a result of
which plus or minus infinity
are going to behave the way you would expect them to,
but we're going to have formalized it.
So our first task is to say, as I said, that the extended real line
is the ordinary real line with two new points ... whatever they are.
Then we have to decide what we're going to
do with these two new points.
And what we have to do is extend our old notions.
So we already have
a total order on the real line, the usual order.
And we just want to extend
that so that minus infinity and plus infinity are included as well.
And that's obvious because just by the notation,
we know what we wanted to do with them.
We say minus infinity's at the left-hand end, plus infinity's at
the right-hand end and the real numbers live in between.
So we simply make the obvious definition
in order to extend less than or equals.
So this is a new less than or equals
extending the old less than or equals.
But we'll still use the same notation.
That why we won't forget that it's the same as it used to be,
if you're actually working with real numbers.
Everything is less than or equal to plus infinity, which is
the same as infinity, and
everything is greater or equal to minus infinity.
All sorts of really nice things happen that
I won't spend too much time on.
Like, you know, when you wanted to have
least upper bounds and greatest
lower bounds, you always had to say,
if you have a non-empty set which is bounded above, or bounded
below, then you can take a supremum
in the real line and so on.
There's no problems like that, in the extended real line it's great.
You just say every subset of the extended real line has got
a supremum, a least upper bound, and
has got an infimum, a greatest lower bound.
Because essentially either you
get the old one that you used to get from the real line,
or you end up with one of the plus or minus infinities,
with a really weird special case of
the empty set because the supremum of the empty set,
the least upper bound of the empty set is
minus infinity and the greatest
lower bound of the empty set is plus infinity.
This is the only set where
the greatest lower bound
is strictly greater than the least upper bound.
More details on that, if you go back
to the measure and integration notes.
We won't need to deal with that ...
the other thing that we did in the measure and
integration notes that we're not going to need in this module
is some very useful things I dealt with there called
the lim inf and the lim sup,
something you can apply to any sequence.
I won't go into that in this module,
but it is very useful if you want to do any further analysis.
So if you do want to go on and
do analysis to any greater depth
I would suggest that you find out that some,
some of the stuff in measure and integration that
we don't cover in this module.
OK! So now, of course, you can define various kinds of intervals.
The old real line is what it was before.
The open interval between minus infinity and infinity.
The extended real line, as
I've already said, is the closed interval between minus
infinity and infinity inclusive.
But most of the time in this module,
we're going to work with the non-negative extended real numbers,
which are extended real numbers between 0 and infinity inclusive.
So most of the time we won't be fussed about minus infinity,
and most of the time we won't we won't be fussed about the
negative real numbers when we're doing the measure theory.
Just occasionally I might mention
something about them. Okay.
Any questions about
the extended real line, or what we've said
so far? All we did so far is the set,
and we're gradually putting some abstract structure on it
so that we know how to manipulate it. So far we've just got an order.
Now we want a notion of convergence.
Now I did this in much greater detail
when I taught measure and integration.
But I'll tell you what you need to know in this module ... actually
well, first of all, I'll give you a way of thinking
about it, it's not the only way of thinking about it.
But the idea is that in
some sense we're going to
treat the closed interval minus infinity, infinity
rather like the closed interval minus one, one.
In other words, I'm going to show you how we can put
a metric on this set which actually will make it isometric to that.
And that will give us a notion of convergence with
respect to which it'll actually be a compact metric space.
We obviously won't extend
the usual metric from the real line, but it
will extend the usual topology.
So what do you have to do? First of all,
we start with a homeomorphism
from the real numbers to (-1,1).
Now there's a lot of these around.
Here's one of them and I don't really care which one you use,
but I would like you to use an order preserving one.
You could flip the order if you want,
that will just make everything a little bit strange.
So I'd prefer you to use an order preserving homeomorphism,
like this one, so the one I'm suggesting
here f(x) is x over (one plus mod x),
and if x is a real number,
then it'll have a modulus
one plus mod x will be slightly bigger than the modulus of x.
When you divide them, you'll get
a real number whose modulus is strictly less than one,
but you can get as close to one as you like.
So a little bit of thought will tell you
that this is doing what it ought to
and taking R to the open interval (-1,1).
Now I'm leaving it to you as a little exercise to think what the,
to find the formula for the
inverse map to make sure it really is one-to-one.
And you can check the order preserving as well.
And there's a homeomorphism.
Well, once you've found the formula for the inverse map,
you just have to observe that that's continuous as well.
This one's obviously continuous.
You can find a formula for the inverse map.
Which is also continuous, then you'll know
it's a bijection. The order preserving
is quite easy, and so on.
I say
any other map with the same properties
would do. Anybody else know another one?
Who can think of something
trigonometric that might ... the first thing you think of
might go, might do pi by 2 ...
minus pi by two and pi by two instead
so you'd have to scale it a bit.
Any suggestions?
[inaudible suggestion from audience]
Ah, excellent!
Okay!
So arctan does
R goes to minus pi by two, pi by two.
That's a homeomorphism.
So you could use
f(x) equals two over pi
(which does your dividing by pi by two) arctan(x),
and that would do just as well.
And that's order preserving homeomorphism from
the real line one to (-1,1). Actually, it doesn't even matter,
it doesn't really have to go to (-1,1),
any open interval would have done
anyway for the rest of the argument,
we just have to move things around a bit,
but I don't really care.
So you've done one of these,
and now, of course, your function f was defined on R,
and it was order preserving, and it's missing the endpoints.
So you just take the endpoints to the endpoints.
So you extend your function by saying we'll
take our left-hand endpoint
to our left-hand, our missing left-hand endpoint,
and our right-hand endpoint to
our missing right-hand endpoint, and you end up with a bijection.
And since it's, we've got no topology yet on the [extended] real line,
so we've now got a bijection, an order preserving bijection
from the extended real line
to [-1,1] which
behaves properly on the real line, but which, you know, we're, so far
it's just a function on the whole thing.
But now you can pull back the metric. Here's the pull-back
metric. I mentioned this before.
You did it in G13MTS probably. If you didn't,
then this is how you pull back a metric.
My distance in R bar, and I've told you
which function f I'm using, but if you want to use
your function, it will be just as good.
It won't change anything else I'm going to
say, so it doesn't matter.
You look at the modulus,
usual modulus in R, of the difference between f(x) and f(y).
So that's the usual distance in R,
or the usual distance in [-1,1],
which is the usual distance in R.
And now you've got a metric space,
isometric to [-1,1],
which makes it a compact metric space.
And of course, because of the way we set it up,
the real line embeds into it
as a sensible subset with its usual topology.
So this extends
the correct topology from the real line to give you a topology and a
metric on ... it doesn't extend the usual
metric, but it does extend the usual topology.
And you've now got a metric and a topology on [-1,1],
and now you've got a notion of convergence,
and that's how you define your convergent sequences.
And I'll say a bit more about this next time,
but if you've got a sequence in R converging
in R with the usual metric,
it still converges where it used, it ought to,
in the extended real line.
All that's happened is we've got some new convergent sequences
converging to plus or minus infinity.
And I will say a little bit more about that, because
we'd like to ... see if you can think for yourself of a
definition, not involving f, that
explains what it means for a
sequence to converge to plus infinity, or to
minus infinity, something with some quantifiers in
like, for all positive real numbers
N, something happens, and try and eliminate the, that
distance and that function that I mentioned
before, to just try and get
hold of what does it mean for a sequence to converge to infinity in
the extended real line,
and you should find out it's rather like
the old notion you had of diverging to infinity,
except that nowadays, in this module,
they're going to converge to infinity instead, which is good.
Okay, I'm going to stop there for today.
We'll finish this off next time, and then we'll
move on to see things
like addition and multiplication in the extended real line.