Let $(X_{n})_{n}$ be independent random variables on $(\Omega, \mathcal{F}, P)$ and define $\tau:=\bigcap_{n \in \mathbb N}\sigma(X_{n},X_{n+1},...)$ and let $Y$ be a real random variable.
Show:
$Y$ is $\tau-$measurable $\iff$ there $\exists c \in \mathbb R$ where $P(Y=c)=1$
I assume that $\Leftarrow$ is easier. Looking at the fact that $\exists c \in \mathbb R$ where $P(Y=c)=1$
$P(Y=c)=P(Y^{-1}(c))=1$ so this would mean that $Y^{-1}(c)\in \tau=\bigcap_{n \in \mathbb N}\sigma(X_{n},X_{n+1},...)$ but then what can I say about any $A \in \mathcal{B}(\mathbb R)$?