Random variables and sigma field Given $Z_1,Z_2,\cdots$ i.i.d with $E|Z_i|<\infty$. $\theta$ is an independent r.v. with finite mean and $Y_i=Z_i+\theta$. 
If we define $F_n=\sigma(Y_1,\cdots,Y_n), F_\infty=\sigma(\cup_n F_n).$ Do we have $\theta\in F_\infty$?
 A: How about: If $E[Z_i] = m$, then
$$
\lim_{n \to \infty} \frac{1}{n}\sum_{k=1}^n (Z_i+\theta) = m + \theta
$$
a.e. so $m+\theta$ is $F_\infty$-measurable.  So is the constant $m$.
A: Concerning GEdgar's followup question; the moments are not necessary.
We replace the sample average and true mean $m$ by the empirical distribution function 
and the  true distribution function $F$.
Without loss
of generality assume that the probability space is $\Omega\times \Omega^\prime$, 
a product space equipped with a product measure $\mathbb{P}\times\mathbb{Q}$. 
For any fixed $c\in\mathbb{R}$ we have (almost surely) 
$${1\over n}\sum_{i=1}^n1_{(-\infty,\, c]}(Z_i(\omega)+\theta(\omega^\prime))
= {1\over n}\sum_{i=1}^n1_{(-\infty,\,c-\theta(\omega^\prime)]}(Z_i(\omega)) 
\to F(c-\theta(\omega^\prime)),\tag1$$ where $F$ is the cumulative distribution function
 of $Z$.  
For each fixed $\omega^\prime$, the convergence in (1) holds for $\mathbb{P}$-almost every $\omega$ by the law of large numbers
 and hence, by Fubini,  holds for $(\mathbb{P}\times\mathbb{Q})$-almost every $(\omega,\omega^\prime)$ on the product space. For $0<u<1$ define $q(u)=\inf(x: F(x)\geq u)$, the quantile function of $F$. It is easy to show that $F(c-\theta(\omega^\prime))\geq u$ if and only if $c-\theta(\omega^\prime)\geq q(u)$.
In particular, this shows that 
$$\{(\omega,\omega^\prime) : c-q(1/2)\geq \theta(\omega^\prime)\}\in {F}_\infty\vee{\cal N}$$
where ${\cal N}$ is the $\sigma$-field of $(\mathbb{P}\times\mathbb{Q})$-trivial sets. 
Since $c$ is arbitrary, we conclude that $\theta\in {F}_\infty\vee{\cal N}.$
