$\mathcal{F}_\infty$-measurablity of a random variable

Question

Let $X$ be a r.v. and $Y_1, Y_2, \dots$ be i.i.d. integrable r.v.'s independent of $X$, with $E[Y_i]=0$.

Here, we consider $Z_i = X + Y_i$, and let $\mathcal{F}_n$ be filtration s.t. $\mathcal{F}_n = \sigma (Z_1, \dots , Z_n)$ and $\mathcal{F}_\infty = \cup_n^\infty \mathcal{F}_n$.

Here, I think $X$ must be $\mathcal{F}_\infty$-measurable, but I'm not sure how to prove it. Any hints? Thanks.

Answer 1

Let $S_n := \frac{1}{n} \sum_{i=1}^n Z_i = \frac{1}{n} \sum_{i=1}^n (X+Y_i) = X + \frac 1n \sum_{i=1}^n Y_i$. Each $S_n$ is $\mathcal F_\infty$ measurable, and $\lim_{n \rightarrow \infty} S_n = X$ a.s. by the strong law of large numbers. Since $X$ is the a.s. limit of a sequence of $\mathcal F_{\infty}$-measurable random variables, $X$ is also $\mathcal F_{\infty}$-measurable.