

Book: "Theory of the Integral" - ypavan
http://banach.univ.gda.pl/e-saks.html

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mahmud
Capinski and Kopp's Measure, Integral and Probability is a good brisk
introduction to the theory as well, albeit a bit hand-wavy at times.

If you're feeling brave, have a shot at Wheeden and Zygmund's "Measure and
Integral" :-)

Zygmund does not do prerequisites, he assumes you know as much as he, so you
will need to do a lot of heavy lifting on your own and augment his works with
other resources. I accidentally bought "Trignometric Series" from Amazon for
$100 once, and when it arrived, I couldn't read anything past the preface. It
sat there on my desk for months; I was messing with signal processing at the
time, so I was reading a lot of materials and going back to it every once in a
while. I kept opening it up and forcing myself, out of guilt. Over time, this
much loathed text became my best friend; whenever "intro" texts became too
chatty, I turned to Zygmund to get the gist of a theorem or a technique. I
started comparing proofs and eventually came to prefer it. Still, was overkill
for someone who isn't an specialist.

[Edit:

I really don't recommend the book in the parent post. It feels a bit antique;
I have had by first brush against analysis via Knopp's cheap Dover books.
Excellent mathematics, yes, but the excessive syntax, weird conventions and
unpronounceable German alphabet made it harder than it needs to be. I
recommend people get set-theoretic intros to analysis; mathematics teaching
has become far more accessible in the last few decades, and nowadays the
yellow Springer-Verlag texts, specially for undergraduates, is doing a fine
job combining rigor with readability. Serge Lang pretty much made mathematics
playful.

After that, move to the hard-cover Princeton Series on Applied Mathematics,
and reach to the blue Cambridge books for absolute rigor, and the company of
masters.]

~~~
noblethrasher
How do you feel about Rudin's Principles of Mathematical Analysis? I've been
slogging my way though that for a while.

~~~
mahmud
It's a beautiful text and kept me company for many months: I have probably
slogged through it slower than you can came out victorious.

For every chapter on Rudin I must have done a whole complimentary text,
though. It's scarce on fluff and every line requires careful attention, but it
will prepare you for analysis better than any ghastly 2000 page beast on
"Calculus".

