Book: "Theory of the Integral"

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.]