Tail $\sigma$-algebra and a sequence of random variables

Question

Let $X_1, X_2, ...$ be random variables. Define $$\mathcal T_n := \sigma(X_{n+1}, X_{n+2}, ...), \quad \mathcal T:= \bigcap_n \mathcal T_n.$$ How can I show that $X_1, X_2,...$ are $\mathcal T$-measurable?

Answer 1

You can't, because in general, they aren't. Take $X_1 \sim\mathcal{N}(0,1)$ and $X_n\equiv 0$ for all $n>1$. Then

$$ \mathcal{T}=\bigcap_{n\ge 1}\sigma(X_{n+1}, X_{n+2}, \ldots)= \bigcap_{n\ge 1} \{\emptyset,\Omega\}=\{\emptyset,\Omega\},$$

and so $X_1$ is clearly not $\mathcal{T}$-measurable.

Answer 2

They are not $\mathcal{T}$-measurable. For example considering a coin toss, if $\Omega=\{H,T\}$, $X_1(H)=1$, $X_1(T)=0$, and $X_2=X_3=\ldots=0$, then $\mathcal{T}=\{\emptyset , \Omega\}$, while the $\sigma$-algebra generated by $X_1$ is $2^\Omega$.