Indicator function, $\sigma$-algebra.

Question

Suppose $\mathcal{F}$ is a collection of real-valued functions on $X$ such that the constant functions are in $\mathcal{F}$ and $f + g$, $fg$ and $cf$ are in $\mathcal{F}$ whenever $f$, $g \in \mathcal{F}$ and $c \in \mathbb{R}$. Suppose $f \in \mathcal{F}$ whenever $f_n \to f$ and each $f_n \in \mathcal{F}$. Define the function$$\chi_A(x) = \begin{cases} 1 & \text{if }x \in A \\ 0 & \text{if }x \notin A.\end{cases}$$How do I see that $\mathcal{A} = \{A \subset X : \chi_A \in \mathcal{F}\}$ is a $\sigma$-algebra?

Answer 1

The only difficult axiom for $\sigma-$algebras to show would be closure under countable unions and intersections. However, this follows from the fact that $\mathcal{F}$ is closed under pointwise limits (i.e. $f_n \in \mathcal{F}\ \forall n \implies \lim f_n = f \in \mathcal{F}$).

To show that this is the case, use the facts that for two sets $A$ and $B$ $A \cup B$ is $\max \{1_A, 1_B \}$, i.e. $$1_{A \cup B} = \max\{1_A, 1_B \}$$ and $$1_{A \cap B} = \min\{1_A, 1_B\}$$ This can then be extended to finitely many sets, and then the closure under pointwise limits will allow us to show that countable unions and intersection of $A_i$ are included in the collection. It may also be useful to make the sequences of sets monotone first, see e.g. An algebra (of sets) is a sigma algebra iff it is a monotone class