What is the relevance of linearity and monotone convergence theorem?

Question

From Williams' Probability w/ Martingales:

5.5 is linearity of integration for non-negative $\Sigma$-measurable functions, and MON is monotone convergence theorem.

What are the relevance of those?

We need to show:

1. $(f\mu)(A) \geq 0 \forall A \in \Sigma$

2. $(f\mu)(\bigcup_n A_n) = \sum_n (f\mu) A_n $ where $A_n$'s are $\Sigma$-measurable and pairwise disjoint.

Here is my proof:

1. $f \ge 0$

$\to f1_A \ge 0 \forall A \in \Sigma$

$\to \int_S f1_A \ge 0 \forall A \in \Sigma$

$\to \int_A f \ge 0 \forall A \in \Sigma$

2. $(f\mu)(\bigcup_n A_n)$

$= \mu (f1_{\bigcup_n A_n})$

$= \int_S (f1_{\bigcup_n A_n}) d\mu$

$= \int_S (f \sum_n 1_{A_n}) d\mu$

$= \int_S (\sum_n f1_{A_n}) d\mu$

$= \sum_n \int_S (f1_{A_n}) d\mu$ ($\because f1_{A_n} \ge 0$ right?)

$= \sum_n \mu(f1_{A_n}) d\mu$

$= \sum_n (f\mu)(A_n) d\mu$

Is that right?

If so: I have a feeling that the $\sum_n \int_S = \int_S \sum_n$ involves MON. Does it? Where are MON or linearity used there?

If not: why? How else can I approach this?

Answer 1

Yes, that equation involves MON.

For $g_n:=\sum_{k=1}^nf_k1_{A_k}$ and $g:=\sum_{k=1}^{\infty}f_k1_{A_k}$ we have $0\leq g_{1}\leq g_{2}\leq\cdots$ and $\lim_{n\to\infty}g_{n}=g$.

So application of the monotone convergence theorem together with linearity gives: $$\int\sum_{k=1}^{\infty}f_{k}1_{A_{k}}d\mu=\int gd\mu\stackrel{\text{MON}}{=}\lim_{n\to\infty}\int g_{n}d\mu=$$$$\lim_{n\to\infty}\int\sum_{k=1}^{n}f_{k}1_{A_{k}}d\mu\stackrel{\text{Linearity}}{=}\lim_{n\to\infty}\sum_{k=1}^{n}\int f_{k}1_{A_{k}}d\mu=\sum_{k=1}^{\infty}\int f_{k}1_{A_{k}}d\mu$$