Part III

(vi) Continuing on, there is the subject
of measure theory. That was from H.
Lebesgue, a student of E. Borel, right, in
France near 1900.

They were correct: They improved on the
Riemann integral. Don't worry: Whenever
the Riemann integral exists, the integral
from Lebesgue's measure theory gives the
same numerical answer.

So, why a new (way of defining the)
integral? Two biggie reasons:

(a) For a lot of theorem proving about
integrals, e.g., differentiation under the
integral sign, clean treatment of what
physics needed from the Dirac delta
function (right, there can be no such
function, but measure theory has a good
answer), definition of an integral of
several variables, interchange of order of
integration in iterated integrals, tying
off some old, loose ends in Fourier
theory, the deep connections between
integration and linear operators, and
more, Lebesgue's work is crucial and
terrific stuff.

(b) Lebesgue's integral is much more
general than the Riemann integral, and
that generality is crucial, especially as
the foundation for probability theory.

Now, for what Lebesgue did:

First, he developed measure theory.
That's essentially just a grown up theory
of area like you have known about since
grade school. E.g., given a set, in the
real numbers, R^n, something more
complicated, or something fully abstract,
the measure of that set is essentially
just its area. With the generalization,
sure, can have some sets with measure
infinity, negative measure, complex valued
measure, etc. But on the real line, with
the usual measure, Lebesgue measure, the
measure of an interval is just its length,
and you already know about that.

But Lebesgue measure is darned general:
It's tricky to show that there is a set of
the reals that is not Lebesgue measurable,
and the usual proof uses the axiom of
choice.

So, for measure theory, there is a
measure space with three things: There
is a space, just some non-empty set,
say, M; there is a collection of
measurable sets, subsets of M called,
say, S; and there is a measure m. Then the
collection of measurable sets S satisfies
the simple, essentially obvious axioms we
would want for area and, thus, is a sigma
algebra of sets. Then for each set A in
S, its measure is the real (or complex)
number m(A). We ask that m have the
properties we want for a good theory of
area. It's just a grown up version of
area. The theory is not trivial; it was
tricky to get all the details just right
so that have a good theory of area, that
is, so that area works like we want it to.

E.g., for the space, can have the set of
real numbers R. For the measurable sets,
have the intervals and, then, all the
other sets needed to have a sigma algebra
(a short proof shows that this definition
is well defined). Then the Lebesgue
measure of an interval is just its length,
and the measure of all the other
measurable sets are what they, then, have
to be (need some theorems and proofs
here).

So, for probability theory, a
probability space is just a measure
space (each point in that space is one
experimental trial); a probability, P,
is just a positive measure with maximum
value 1; an event A is just one of the
measurable sets; and the probability of
A is just its measure P(A). What we want
for probability is already so close to a
theory of area that, really, we have
little choice but just to follow what
Lebesgue did. That's what A. Kolmogorov
observed in his 1933 paper.

Second, with the foundation of measure
theory, Lebesgue defined the Lebesgue
integral.

So, what is being integrated is (usually)
a function taking real or complex values.
The domain of the function is a measure
space.

Then for the integral, say, in the
case of a real valued function,
we partition on the Y axis,
that is, in the range of the function
instead of in its domain. So, we don't
have to do Riemann-like partitions of the
domain of the function and, thus, the
domain can be much more general.

As the first step, we only integrate
functions that are >= 0, and we do that by
approximating, right, again with
essentially rectangles, only from below
the function, not both above and below as
for Riemann. Here the domain of the
function can be the whole space, e.g., the
whole real line. We don't care about
either continuity or compactness.

For a function that is both positive and
negative, we multiply the negative part by
-1 and integrate the two parts separately.
If at least one of the two results is not
infinity, then we subtract, and that's the
integral.

A random variable is just such a
function, and its expectation is just
its integral.

Summary

Don't feel like the Lone Ranger; not
everyone knows this stuff. E.g., from all
I've been able to see from quantum
mechanics, the wave functions are
differentiable. Then I'm told that the
wave functions, wondrous, are also
continuous. From baby Rudin, of course
they are continuous; every differentiable
function is continuous! Then I'm told
that the wave functions form a Hilbert
space. Well, I can see that they can be
points in a Hilbert space, but they can't
form a Hilbert space because the
continuous functions won't be complete.

In elementary probability, it is common,
e.g., for finding the expectation of a
Gaussian random variable, to integrate
over the whole real line. Tilt: Baby
Rudin defines the Riemann integral only
over compact sets. So, can use an
improper integral, and then they want to
differentiate under the integral sign.
Tilt again -- the needed theorems are not
so easy to find. So, really, this common
stuff, in probability, physics, etc. of
integrating over the whole real line or
over all of R^n is using the Lebesgue
theory where have clean theorems for such
things.

That's your overview and your work for the
weekend. You are now permitted two beers
and half a pizza but only if you have
someone you like a lot for the rest of the
pizza and some more beer. Can substitute
Chianti for the beer. But no more math
permitted; maybe a movie, but not math!