Proof of Monotone Convergence Theorem in Cinlar's "Probability and Stochastics"

Question

In Cinlar's book "Probability and Stochastics", the Monotone Convergence Theorem is stated as follows:

Let $(f_n)$ be an increasing sequence in $\mathcal{E}_+$. Then, $\mu(\lim f_n) = \lim \mu f_n$.

($\mathcal{E}_+$ is the set of positive measures for some measurable space $(E, \mathcal{E})$.)

The proof then proceeds as follows:

[...]

Fix $b$ in $R_+$ and $B$ in $\mathcal{E}$. Suppose that $f(x) > b$ for every x in the set $B$.

[...]

$f_n 1_B \geq f_n 1_{B_n} \geq b 1_{B_n}$

I see that $f_n 1_B \geq f_n 1_{B_n}$ holds. My question is: Why does $f_n 1_{B_n} \geq b 1_{B_n}$ hold? After all, just because $(f_n)$ increases to some $f$ that's $\geq$ than $b$ for domain $B$, an individual $f_n$ may still be $< b$ for some $x \in B$?

I hope that I extracted all relevant context from the text for this question. The full proof can be found in Cinlar's book "Probability and Stochastics", chapter 1, section 4.

Answer 1

I hope that I extracted all relevant context from the text for this question.

I didn't :). The crucial detail I missed on the first read is the definition

$$B_n = B \cap \{f_n>b\}$$ where $\{f_n>b\}$ is defined as $\{x : f_n(x) > b\}$. Now, clearly $f_n 1_{B_n} \geq b 1_{B_n}$ holds, because $1_{B_n}(x)$ is $1$ only if $f_n(x) > b$.