measure-theory

I am reading the proof that the pointwise limit of measurable functions is measurable in Axler's Measure, Integration & Real Analysis, which states: Theorem 2.48 (Limit of $\mathcal{S}$-...

I am currently self-studying Measure, Integration & Real Analysis by Axler, and I am struggling to understand the motivation behind the definition of a measurable function. The definition I am ...

Now that we've covered a few ways we can understand the Integral and the basics of Measure Theory and Probability Spaces we are ready to put all of that together to introduce a very exciting topic: the Lebesgue Integral. Like everything else we've discussed recently, this topic is typically reserved for advanced study, but the basic idea is relatively simple to understand and very practical for d…

There are a number of ways to construct the real numbers , for instance as the metric completion of (thus, is defined as the set of Cauchy sequences of rationals, modulo Cauchy equivalence); as the space of Dedekind cuts on the rationals ; as the space of quasimorphisms on the integers, quotiented by bounded functions. […]