The following is excercise 5.5.2 from Durrett's Probability Theory and Examples.
Let $Z_{1}, Z_{2}, \ldots, Z_{n}$ be i.i.d. with $\mathbb{E}|Z_{i}|<\infty$, let $\theta$ be be an independent random variable with finite mean, and let $Y_{i}=Z_{i} + \theta$. Show that $\mathbb{E}(\theta| Y_{1}, Y_{2}, \ldots, Y_{n}) \to \theta$ a.s.
I have started by using theorem 5.5.7 in Durrett to show that, if $\mathcal{F}_{n} = \sigma(Y_{1}, \ldots, Y_{n})$, then we have $$\mathbb{E}(\theta | Y_{1}, Y_{2}, \ldots, Y_{n}) = \mathbb{E}(\theta | \mathcal{F}_{n}) \to \mathbb{E}(\theta | \mathcal{F}_{\infty})$$ a.s. and in $L^{1}$, where $\mathcal{F}_{\infty} = \sigma\left( \bigcup_{i\geq 1} Y_{i}\right)$. Now if I knew that $\theta$ was measurable w.r.t. $\mathcal{F}_{\infty}$ then I would have that $\mathbb{E}(\theta | \mathcal{F}_{\infty}) = \theta$, but sadly I do not know that this is true...
I am pretty much stuck here. I do not know how to incorporate that the $Z_{i}$'s are i.i.d. with finite mean, or that $\theta$ is independent with finite mean.
I am honestly so bad at this stuff. Any help is appreciated.