Is it known whether the following version of the strong law of large numbers holds?

For each $k\in\mathbb{N}$, let $\Omega_k$ be a finite set and $\mu_k$ be a probability measure on $\Omega_k$. Let $(X_k)_{k\in\mathbb{N}}$ be a sequence of random variables, $X_k: \Omega_k \to [0,+\infty)$. Suppose in addition that $\sup_k \int X_k\,d\mu_k < \infty$.

Then with $\prod_{k=1}^\infty \mu_k$-probability one, we have $\lim_{N\to\infty}\frac{1}{N} \sum_{k=1}^N (X_k - \int X_k\,d\mu_k) = 0$.