Part II

Starting there, as you go on in real
analysis, you will place much less
emphasis on continuity. The main work
will replacing the Riemann integral by
something that does much better on the
edge or pathological cases.

Well, the Riemann integral was, right, a
lot about area. Well, it's actually not
a really good theory of area -- doesn't
have all the nice properties we would wish
for. So, that view of area gets
polished up and replaced by measure
theory. Next, that theory of area is used
to define the Lebesgue integral, which is
much nicer than the Riemann integral,
e.g., gets rid of the assumptions of a
compact set and continuity.

Lebesgue was a student of E. Borel and did
his work near 1900. You will see the
Heine-Borel theorem in Baby Rudin.

In simplest terms, the Lebesgue idea is to
partition on the Y axis instead of the X
axis like the Riemann integral does. Just
why that works so well is cute. It also
frees up the domain of the function to be
much more general, and in 1933 A.
Kolmogorov used that fact, finally, to
give a good foundation for probability
theory, the one accepted now in
essentially all advanced work in
probability, stochastic processes, and
mathematical statistics.

Likely the nicest book to read on real
analysis is H. Royden, Real Analysis.
But also, later, read the first, real half
of W. Rudin, Real and Complex Analysis.

Note: For real analysis, complex valued
functions of a real variable are
important; functions of a complex variable
are quite different and not important;
Rudin has some deep reasons to disagree
with me. Rudin is no doubt correct; for
students trying to learn, I'm correct!

Once you have the main ideas in mind, and
maybe what I've given here will be enough,
Rudin is more succinct and nicer to read
than Royden. But read them both, Royden
first. Go through Royden quickly since
will do it all again in Rudin.

In Royden, notice his Littlewood's Three
Principles as a cute view of what is going
on in measure theory and the Lebesgue
integral. Royden also has some exercises
on upper and lower semi-continuity --
glance at those and get the main idea for
the remaining connection with continuity.
Notice the nice treatment of
differentiation -- the connection between
integration and differentiation is weaker
for the Lebesgue integral because the
integral is so much more general. Then
notice the Carathéodory extension result
-- that is the key to defining measures on
the real line, in particular, Lebesgue
measure.

Spend at least a weekend in

John C.\ Oxtoby, {\it Measure and
Category:\ \ A Survey of the Analogies
between Topological and Measure Spaces,\/}
ISBN 3-540-05349-2, Springer-Verlag,
Berlin, 1971.\ \

An amazing weekend.

Keep at hand

Bernard R.\ Gelbaum and John M.\ H.\
Olmsted, {\it Counterexamples in
Analysis,\/} Holden-Day, San Francisco,
1964.\ \

Notice that for the Lebesgue integral,
there is one, central proof technique in
just four steps: Prove the theorem for a
function that is just a positive constant
defined on a set, say, just a box. By
linearity, generalize to finite sums of
such functions. Then by monotone
continuity (see early on the monotone
convergence theorem), prove the result for
all non-negative functions. Last by
linearity again, get the result for all
integrable functions.

Okay, for integrable, there is a cute
approach, trick: Define the integral for
non-negative functions, even ones defined
on the whole real line or all of R^n (for
the set of real number R and a positive
integer n) or an abstract measure space.
Okay, easily enough, that integral can
have value positive infinity (the Riemann
integral gets sick here). Then do the
same for a function that is all <= 0. Then
given a function, write it as the sum of
its positive part and its negative part.
Integrate those two parts separately.
Now, if at least one of those two
integrals is finite, then add them for the
integral of the given function. So, we
are okay in all cases except where the
positive part integrates to positive
infinity and the negative part integrates
to negative infinity -- we don't want to
subtract infinities because permitting
that ruins the laws we want for
arithmetic.

For this four step proof technique, can
knock off a really nice version of
Fubini's theorem, that is, interchange of
order of integration.

The Lebesgue integral with the dominated
convergence theorem, which will see early
on, give nice versions of differentiation
under the integral sign.

Note: Commonly in physics, engineering,
probability theory, etc. we write
integrals over the whole real line. Alas,
really, for Riemann integration, the
appropriate theorems are rarely presented;
they are not in Baby Rudin. Really, the
Lebesgue integral is needed. So, you've
been needing the Lebesgue integral for a
long time.

Then get to apply the Lebesgue integral to
the powerful Radon-Nikodym theorem (Rudin
gives von Neumann's cute proof based,
amazingly, in part on just polynomials),
get the main, important duality theorems
(the Lebesgue integral is the main linear
operator!), Banach spaces, Hilbert spaces,
and the Fourier integral.

Cover that and can claim you have a good
start on real analysis. You will also
have apparently so far the only good
background for probability, stochastic
processes, and mathematical statistics.
E.g., the Radon-Nikodym theorem is crucial
for conditional expectation, Markov
processes, martingales (amazing things
with one of the strongest inequalities in
math and a super short proof of the strong
law of large numbers), ergodic theory,
sufficiency in statistics, a quite general
proof of the Neyman-Pearson result in
statistical hypothesis testing, and more,
e.g., the role of Brownian motion in
potential theory, stochastic optimal
control, ....

Go for it!