Suppose we have i.i.d. random variables $\{X_1, X_2, \cdots\}$. And for each $X_i$, we have i.i.d. random variables $\{Y_{i, 1}, Y_{i, 2}, \cdots\}$ that are conditioned on $X_i$. Generally, $P(Y|X_i)\neq P(Y|X_k)$ if $i\neq k$. Additionally, we assume $Y_{i,j}$ are bounded or even binary.
Now we have a double array $\{Y_{i,j}:i\geq 1,j\geq 1\}$. For each $i$, we define $$S_{i,n}=\frac{1}{n}\sum_{j=1}^n Y_{i,j}.$$ By the strong law of large numbers, we have $$S_{i,n}\to E[Y|X_i] \ \ \ a.s.\ \ \ n\to \infty.$$
My question is about the convergence of the mean of the double array: $$\frac{1}{n}\sum_{i=1}^n S_{i,n} \overset{?}{\to} \mu = E[Y] = E[E[Y|X]], \ \ n\to \infty.$$
If regard $Z_i=E[Y|X_i]$ as a random variable, we have $$\frac{1}{n}\sum_{i=1}^n Z_i \overset{}{\to} \mu \ \ a.s. \ \ n\to \infty.$$ However, I found it may be incorrect to derive directly $$\frac{1}{n}\sum_{i=1}^n S_{i,n}\to\frac{1}{n}\sum_{i=1}^n Z_i\to \mu \ \ \ a.s.\ \ n\to \infty.$$