Sigma algebra problem

Question

Suppose that $\mathcal{A}$ is a sigma algebra and $F:=\left\{A\in \mathcal{A}:\mathbb{P}(A)=0 \ \text{or} \ \mathbb{P}(A)=1 \right\}$ , where $\mathbb{P}$ is a probability measure.I have to prove that $F$ is sigma algebra.I proved the complements property and that $\Omega \in F$.But how should I prove that if $\left\{A_{n}\right\}_{n\in\mathbb{N}}\subset F \Rightarrow \cup_{n\in \mathbb{N}}A_{n}\in F$??

Answer 1

Suppose that $\{A_n\}_{n=1}^{\infty}$ is a sequence of elements of $\mathcal{F}$. If $\mathbb{P}(A_n)=0$ for all $n$, then $$ \mathbb{P}\Big(\bigcup_{n=1}^{\infty}A_n\Big)\leq \sum_{n=1}^{\infty}\mathbb{P}(A_n)=0$$ while if $\mathbb{P}(A_k)=1$ for some $k$ then $$\mathbb{P}\Big(\bigcup_{n=1}^{\infty}A_n\Big)\geq \mathbb{P}(A_k)=1$$ and since $\mathbb{P}$ is a probability measure it follows that $$\mathbb{P}\Big(\bigcup_{n=1}^{\infty}A_n\Big)=1$$ In either case we conclude that $\cup_{n=1}^{\infty}A_n\in\mathcal{F}$.