Questions about integrals over subsets

Question

From Williams' Probability w/ Martingales:

1. What exactly is meant by $f|_A$? If we have $f: \mathbb{R} \to \mathbb{R}$ and A = [0,1], does that mean $f|_A: [0,1] \to \mathbb{R}$?

2. Is it that $f|_A = f1_A$? It seems that $f, 1_A$ and $f1_a: \mathbb{R} \to \mathbb{R}$ while $f|_A: [0,1] \to \mathbb{R}$ so I was thinking that $f|_A = f1_A$ doesn't make sense in the first place?

3. Should B belong to $\Sigma$ and $\Sigma_A$? I have a feeling this was mistakenly omitted or omitted because it is thought to be understood, if I am right. Iirc, $A \in \Sigma$ and $B \subseteq A$ do not imply $B \in \Sigma$

Answer 1

1. Yes. $f|_A$ is the restriction of $f$ to $A$.

2 . Technically they're not the same, $f|_A$ has domain $A$, while $f 1_A$ has domain $\mathbb{R}$. But they might as well be the same. If $C \subseteq \mathbb{R}$, and $g$ is a function from $C$ to $\mathbb{R}$, there's no harm in thinking of $g$ as a function from $\mathbb{R}$ to $\mathbb{R}$ which takes the value $0$ outside of $C$.

3 . Yes, you're right. I think $B$ has to be in $\Sigma_A$ for what he's saying to make sense.