sigma algebra of all 0 and 1 events

Question

Let $\mathcal{F}$ be a sigma-algebra over a set $\Omega$. Now let

$$ \mathcal{G} := \{E \in \mathcal{F} : \mathbb{P}(E) = 0 ~\text{or}~1\}. $$

I have to show that $\mathcal{G}$ is a sigma algebra.

The only difficult is to show that if $E_{1},E_{2},\cdots \in \mathcal{G}$ then $\bigcup_{k \geq 1}E_{k} \in \mathcal{G}$.

My first attempt was to use $\mathbb{P}\left(\bigcup_{k \geq 1}E_{k}\right)$ $\leq$ $\sum_{k\geq 1}\mathbb{P}(E_{k})$. Now I'm stuck.

Does anyone have any idea?

Thanks!

Answer 1

$\renewcommand{\Pr}{\mathbb{P}}$ $\renewcommand{\c}[1]{\left(#1\right)}$ Assume without loss of generality that you have $E_1,E_2,\dots,F_1,F_2,\dots\in\mathcal{G}$ s.t. $\mathbb{P}\c{E_i}=1$ and $\Pr\c{F_i}=0$ and prove that: $\bigcup E_i, \bigcup F_i \in \mathcal{G}$.

And all you have left to prove now is that if $E,F\in\mathcal{G}$ s.t. $\Pr\c{E}=1$ and $\Pr\c{F}=0$ then $E\cup F\in\mathcal{G}$.