Is a compact set with respect to the product topology measurable?

Question

Let $X_1,...,X_n$ be metrizable spaces.

Let $\Sigma_i$ be a $\sigma$-algebra on $X_i$ such that every compact subset of $X_i$ is an element of $\Sigma_i$, for each $i$.

Then, is every compact subset of $\prod_{i=1}^n X_i$ an element of $\otimes_{i=1}^n \Sigma_i$?