Welcome back to in-person instruction. I will continue type in here as a way to prepare for class.

After a long toil of last two weeks, we have established the existence of measurable sets and Lebesgue measure on $\R^n$. We know open sets and closed sets are measurable, and countable operations won't take us away from measurable sets. The Lebesgue measure on measurable sets satisfies all the intuitive properties that you wish it has.

Next, we will consider a measurable function. What is a measurable function? Just as in topology, given two topological spaces $X, Y$, we consider continuous function $f: X \to Y$, such that pre-image of open sets are required to be open. Here, if $X, Y$ are 'measure space', i.e equipped with a collection of measurable subsets $\mathcal{M}_X$ and $\mathcal{M}_X$, we want