Measure Theory (2 Volume Set) - Comprehensive Guide for Advanced Mathematics & Real Analysis | Ideal for Graduate Students & Researchers

I have spent time with both volumes of this two volume work. Bogachev presents everything in the language of measure theory, and thus talks about measurable functions rather than random variables. The language of probability theory is slick like an Apple computer, but it hides some inner workings that stare you in the face when you work using the language of measure theory. A probability votary might assert that one shouldn't think about the objects that do not explicitly appear in probability theory. Bogachev covers both more things than other books I have seen and gives more detailed proofs. Many results that are part of mathematical folklore are proved here, like consequences and equivalent statements of equi-integrability/uniform integrability and the Dunford-Pettis theorem; relations between different types of convergence; and the symmetric difference metric on the quotient of a sigma-algebra by the null sets. Also, the theorems are stated with quite general conditions, for example not assuming that we are working with a probability measure when it is enough to work with a sigma-finite measure and not assuming that a metric space is Polish when it is enough that it be separable. In the second volume there is a good exposition of the Borel sigma-algebra of a topological space. Bogachev defines the product of measure spaces and proves the Kolmogorov consistency theorem, which has an especially tractable form for products of Polish spaces.The first volume also has substantial material on differentiability of functions. It has the best proof of the Denjoy-Young-Saks theorem that I've found in a textbook, that the set of points where an arbitrary function on an interval is differentiable is a Borel set. The Dini derivatives used as tools for this.