Is $\sigma (X_\infty) = \mathscr{T}_\infty$ true?

Question

Let $X_n$ be random variables which converges to $X_\infty$. Let $\mathscr{T}_n = \sigma(X_{n+1},X_{n+2},...)\searrow\mathscr{T}_\infty$. Then is it true that $\sigma(X_\infty) = \mathscr{T}_\infty$? It's clear that $\sigma(X_\infty) \subset \mathscr{T}_\infty$, but I can't quite see whether the converse is true or not.

Answer 1

On $(0,1)$ with Lebesgue measure let $X_n=\frac 1 n I_{(0,\frac 1 2 )}$. Then $X_{\infty} =0$ and $\{\sum_1^{\infty} X_n <\infty\}$ is an event in $\mathcal T_{\infty}$ but this event is exactly $[\frac 1 2 ,1)$. which is not in $\sigma (X_{\infty})$ because this event has probability 1/2 (and every event in $\sigma (X_{\infty})$ has probability 0 or 1).