[proofplan]
The joint law of independent variables is the product of the marginal laws. Integrating a product function against a product measure factors into the product of one-dimensional integrals.
[/proofplan]
[step:Use the product law]
Let $\mu_i$ be the law of $X_i$. Independence gives that the law of $(X_1,\ldots,X_n)$ is $\mu_1\otimes\cdots\otimes\mu_n$. Therefore
\begin{align*}
\mathbb E\left[\prod_{i=1}^n h_i(X_i)\right]
=\int_{E_1\times\cdots\times E_n}\prod_{i=1}^n h_i(x_i)\,d(\mu_1\otimes\cdots\otimes\mu_n)(x_1,\ldots,x_n).
\end{align*}
[/step]
[step:Factor the integral]
By Tonelli's theorem for nonnegative parts and then the assumed integrability for the signed functions,
\begin{align*}
\int_{E_1\times\cdots\times E_n}\prod_{i=1}^n h_i(x_i)\,d(\mu_1\otimes\cdots\otimes\mu_n)
=\prod_{i=1}^n\int_{E_i}h_i(x_i)\,d\mu_i(x_i).
\end{align*}
Since
\begin{align*}
\int_{E_i}h_i\,d\mu_i=\mathbb E[h_i(X_i)],
\end{align*}
the desired identity follows.
[/step]
