Halmos Measure Theory is a good
prerequisite to Neveu. Rudin,
Principles is a bit too little.
Instead, the first half, the real
half of Rudin's Real and Complex
Analysis is a good prerequisite.
So, is Royden's Real Analysis.

Neveu is elegant beyond belief,
but Breiman, Probability, the SIAM
book, available in paperback, is
darned good, usually easier
than Neveu, less elegant, closer to
applications, and without some of the
special Tulcea material in the back
of Neveu. K. L. Chung also has a
good, comparable book. Even if want
Neveu to be your main probability book,
which is fine, likely you should have
alternative treatments.

Of course, there is Loeve, Probability
-- written in English but somehow
sounding like French. It has a lot,
a little too much, but I liked the
topics I studied in it. It turns out,
Neveu and Breiman were both Loeve
students.

Halmos, Measure Theory,
is darned fun to read:
It has the three series theorem
and a famous exercise on regular
conditional probabilities.

I learned the stuff from a course
by A. Karr, a star student of
E. Cinlar. Karr's course was the
best course of any kind I ever took
in school. Powerful material,
beautifully presented, each day it
was a shame to erase the board.

The exercises in Neveu are usually
harder than the ones in Halmos,
Breiman, and Chung.

Neveu makes probability a crown jewel
of civilization.

The summer after Karr's course,
I sat in the library for six weeks
and walked out with a 50 page
manuscript that was all the research
and the first draft of my dissertation.
Net, probability at the level of Neveu
is darned powerful stuff, makes a lot
in research, and research for applications,
really easy -- that is, you really
know just what the heck you are doing
and can knock off new results
having fun sitting in bed next to
your wife while she watches TV
(warning -- not gender neutral!).

What I've outlined is sometimes
just called graduate probability.
The biggest difference is that the
whole subject makes daily use
of measure theory.

I don't know how much you need in
probability before starting on
graduate probability. In my case,
graduate probability was my
first serious study of probability,
and I never felt that I was not prepared.

But in my career I'd done a lot of
practical work in both probability
and statistics -- e.g., multivariate
statistics, hypothesis testing,
stochastic processes, digital filtering,
the fast Fourier transform, beam forming
(a case of antenna theory),
power spectral estimation (US Navy
sonar type stuff), how to get the
central limit theorem out of
digital filtering,
and more, random number generation, etc.
That
work was plenty of intuitive background
for graduate probability.

But in much of that work I
was struggling due to what, really,
at that level, is commonly
weak basic knowledge of probability.
So, after those struggles, seeing
graduate probability be all
clean and powerful was great.

I can't advise on just how much
elementary probability you might
need to have enough intuition
to be comfortable with graduate
probability. I will say, you do
need both the intuitive experience
and also the solid math.

I feel sorry for people who work
in prob/stat without a background
in grad prob: The elementary
stuff is too often just confused
from poor understanding from
a poor background.

The sources I mentioned above were really
the first sources from which I did
any real study. Net, the elementary
material of prob/stat is really too simple
to be taken very seriously. So, for your
first serious effort, just go for
graduate probability from the sources
above.

The Neveu, etc., material is much of the
foundation for the secret sauce
of my startup.