Understanding Corollary 12 on pg.83 in Royden 4th edition.

Question

This question is based on my difficulties in understanding Corollary 12 on pg.83 of Royden's Real Analysis (4th edition):

• How it is a corollary of monotone convergence theorem?

• What is the main idea of this corollary and how it increases our freedom in the passage of the limit under the integral sign?

• What does it mean that $f = \sum_{n=1}^{\infty} u_{n}$ pointwise a.e. on $E$?

Answer 1

We have that $\sum_n u_n=u$ if $s_n(x)=u_1(x)+...+u_n(x)$ converges pointwise to $u(x)$ outside of a set of measure zero.

In real analysis,with the Riemman integral,in order to interchange the integral and summation,we must have uniform convergence,which is a very strong mode of convergence.

But with the Lebesgue integral we just want the functions to be non-negative.

This is a corollary of monotone convergence because $s_n(x)$ is non-negative and increasing and and $\int s_n=\sum_{k=1}^n\int u_n$

Answer 2

Regarding your second question, $f = \sum_{n=1}^{\infty} u_{n}$ pointwise a.e. on $E$ iff there is a set $F \subseteq E$ and $\lambda(E \backslash F) = 0$ (we sometimes say $F$ is conull in $\mathcal{B}(E)$) and for all $x \in F$, $f(x) = \sum_{n=1}^{\infty} u_{n}(x)$.

Regarding 1, consider that you have just shown that the sum of the integral is the integral of the sum in some cases. A lot of the time, mathematics is about working out when you can swap things around like that. Try framing monotone convergence in that way. Think about other things you might want to swap around. When is it okay to do that?