Okay. Let's have a look at the start of the next section.
We've got sigma algebras.
We've got at least one interesting sigma algebra, the Borel sets.
And now I'm going to give you the notion of a measurable space,
and measurable sets, and then the notion of a measure,
which is what we're going to be measuring the size of our sets with.
So the ... what's our ... what's the idea of all this?
Well, we've got some informal notions
in our head of length, area, volume.
We've got a pretty good idea, we've got a pretty good intuitive idea
of what area and volume should be.
But ... it's not obvious whether we can
actually assign an area or a volume to every set in a sensible way.
And it turns out there's a problem.
And in one dimension it's
a bit less obvious. I mean, area if you've got
several different sets and
you take their union,
and they don't overlap, then you think the areas should add up.
And, you know, if you've got two disjoint disks,
then you want to add up the two areas of those disks
in order to get the area of their union.
And, if you're lucky,
you can even do this with countable unions.
And it turns out that for sensible sets you can.
But there are some sets that are so
horrible that there's no sensible way to
assign a volume to them or an area to them or, in one dimension,
some equivalent: total length.
And we're going to see that there are some problems.
And so that means that you don't
expect to be able to find
a measure of size that necessarily works for everything.
In which case you find, okay,
but at least for all sensible sets we'll be okay.
'All sensible sets' will include the Borel sets,
which will be almost everything you're ever going to meet.
So as long as we can be sure,
that there is some suitable notion of area, volume, length,
whatever, that works on your Borel sets, that will be
enough to get you going, and to cope with
anything that you're ever going to meet in the real world.
And it's only going to be weird,
pathological counterexamples that you're
never actually going to come across,
whatever 'coming across' means in
Pure Mathematics, that ... that will give you trouble.
So we start with the notion of a measurable space.
Just as the discrete topology is
this very extreme example where
every set is open, but you don't normally want to use it.
With a measurable space, we'll have a notion that
the sets in your sigma algebra will be 'measurable'.
And that will maybe have lots of sets in,
but you don't expect that every set
is going to be measurable, except in extreme cases like
the, you know, just like the discrete topology.
Well, actually it will be the discrete topology!
So ... a measurable space just means
you've got a set and a sigma algebra on it.
This is just like
when you were in a topology, it was just a set with a topology on it.
But it's a different kind of collection of sets, it's a sigma algebra.
And then, just like with a topology
you said the sets that were in tau,
or whatever it was you were calling it, were called the open sets.
Now we say the sets
that are in the sigma algebra are called the measurable sets.
If you have to,
if you've got more than one sigma algebra around,
you might call them the F-measurable sets.
And I might occasionally say
'Borel measurable sets' to remind you
I'm working with the Borel sets in the real line,
that sort of thing.
But typically we'll just have
a measurable space with some measurable sets.
And ... I may not
even have to tell you the name of the sigma algebra some of the time.
So you've got a measurable space,
it's got some sigma algebra on it.
So it's a set, it's got a sigma algebra,
that makes it a measurable space. And typically you
might think of taking the real line, putting the Borel sets on it,
making that into a measurable space that way.
We'll sometimes use a bigger sigma algebra,
but that's quite common for what we're going to look at.
And that's what we say here:
by default, we use the Borel sets on the real line,
but just occasionally we might use some bigger sigma algebra.
And I'll mention the Lebesgue measurable sets a bit later,
or you ... and you can also find out more about
the Lebesgue measurable sets
in ... in books, of course,
or in the notes from that old module of mine,
G13MIN Measure and Integration.
So now ... we're going to try to measure the size of our sets.
And in order to do that,
we're going to introduce the notion of a measure.
So what is a measure?
Actually, there's more than one kind of measure,
but we're going to work with non-negative measures,
which are often called positive measures for some reason.
But they definitely can take the value 0.
But I have seen people who call them non-negative measures,
so maybe that's not a bad name for them.
We won't look at complex measures or
more general signed, real-valued measures, things like that,
... in this module.
But they are available, and they are important in ...
in Functional Analysis, as well as other areas.
So ... I'll just give you the definition of what a measure is
and we'll look at lots of examples next time
to get the idea of, of what they can do.
So ... you might think that part one is redundant,
I've made that mistake once in the past,
but it's a long time ago now.
You do need to insist ...
in addition to condition (2) ... you can't deduce (1) from (2).
You almost can, but you can't quite!
So you need to specially
insist that the measure of the empty set is 0.
The second one says that if you've
got a sequence of pairwise disjoint sets,
that means they don't overlap,
I'll remind you of that next time.
But no two of the sets in your collection should overlap at all:
we may need ... say ...
no set in this collection meets any of the others,
that's what pairwise disjoint means,
then I want the measures to add up correctly.
And this is the sort of thing we want a notion of size to do.
We can't hope for much more than this because, suppose
you try to go for uncountable unions then,
well, suppose you were working with subsets of
the real line and you wanted to talk about length,
then the length of a point will be 0.
Or supposing you're in R squared
the length of ... the area of a point is going to be 0.
Every set is a union of points, usually uncountable,
but even an uncountable sum of zeros would be 0.
So we don't want to go past countable ... sums.
So if we're going to have points having area 0,
then we're going to hope that the area
of a countable set of points is still 0.
And it will be, once we've explained something about Lebesgue measure.
Well I'm mostly, I'm going to be working in one dimension
rather than two. I'll just occasionally go back and talk about
area because it's a bit more intuitive ... Okay,
and then if you got one of these,
and we can see lots of examples next time,
and you put it together with your set and your sigma algebra,
so you had a measurable space, and now you've got a measure on it.
And so you get a measure space,
which is a combination of a measurable space with a measure.
And typically you might be working with the real line,
the Borel sets and
some really nice measure
like Lebesgue measure that we're going to look at in a bit,
which will measure the length of sets,
the total length of sets.
So if you've got two disjoint intervals,
then it's going to add up the lengths of those intervals.
If you've got a disjoint union
of a sequence of intervals, Lebesgue measure
will add up the lengths of those intervals.
And then we'll go on and measure the lengths of every open set.
Well, an open set is a union of a sequence of intervals.
So Lebesgue measure will give every open set in R
exactly the right length, by adding up the lengths of those intervals.
But it will go further and will
give a notion of length to
far more complicated sets, where it's not obvious.
Since the length of a point is going to be 0,
the rational numbers are countable,
the length, total length of the rational numbers is going to be 0,
when you use Lebesgue measure. The irrational numbers,
just like the whole real line, will have infinite length.
So there's far more irrational numbers than rational numbers.
We already knew that the rational numbers were countable
and the irrationals were uncountable.
But this really puts a bit more
information into it.
The rational numbers.
have now got measure 0,
and the irrational numbers have got what's called 'full measure'.
Whatever interval you look in,
there's a full measure set of irrationals.
And the, and the rationals have just got measure 0 there.
So there really are ...
there's far more irrationals than rationals,
even though if you try
and sketch either set on the real number line
all you get is lots of dots apparently ...,
the rationals are some, are sort of
negligible when we come
to do any integration using Lebesgue theory.
And the irrationals are as good as the whole thing.
If you are looking at the irrationals in an interval,
and you decide you're going to integrate over that,
that's going to be just as good
as integrating over the whole of the interval,
when it comes to Lebesgue integration theory.
We'll only have a limited amount of time
to look at this theory in this module.
I'll refer you back to measure and integration for
more detail and more information and more results.
But I will give you at least a bit of a fly through
some of what's going on and how powerful this theory can be,
but referring you to reading for more information.
Ok! We'd better stop there, we've overrun.