My friend and former colleague Christer Samuelsson sent me a present a while back (if you send me presents, you may be mentioned here too, though I don’t have mugs for you):
- Kolmogorov, A. N. and S. V. Fomin; translated and adapted by Richard Silverman. 1970. Introductory Real Analysis. Dover.
Yes, it’s written by the Kolmogorov, the man who invented (discovered, if you’re a Platonist) a consistent basis for probability using measure theory. Not content with laying the foundations for probability, Kolmogorov also kicked off the study of algorithmic information theory through what later became known as Kolmogorov complexity, and I can also highly recommend Li and Vitanyi’s book on that subject, though I’ve only read the introductory bits.
You can check out the contents by following the Amazon link above. I’ll summarize by saying it covers the basics, from set theory to metric topology, including a long discussion of linear spaces and operators, a chapter on measure, and sections on Lebesgue integration in the first integration chapter and the Stieltjes generaliation later (really cool, in that it lets you merge integration and summation).
It presupposes the requisite “mathematical sophistication”, which really means you’ll be hard pressed to read more than a page every 10 or 15 minutes. Seriously, though, if you didn’t understand calc in high school (or college), this probably isn’t the book for you. In fact, if you haven’t studied higher math at all, this probably isn’t the place to start. Having said that, I hated calc in school, and only felt I understood it after doing analysis and topology, which cranked the level of abstraction beyond and Euclidean distances. It’d help to have some background in abstract algebra and set theory, too, though everything is explained clearly (though quickly) from first principles.
I don’t know whether to thank Kolmogorov, his co-author, or his translator, but this book is wonderfully clear. The examples are direct, the notation’s clear, and it’s fairly easy to use modularly (e.g. just to look up what a measure or σ-algebra is).