Let $X_1,...,X_k$ be independent random variables on the measure space $(\Omega,\mathcal{A},\mathbb{P})$ and $h$ a measurable and bounded function from $\mathbb{R}^k$ to $\mathbb{R}$. We denote $\mathbb{P}_{X_i} = \mathbb{P} \circ X_i^{-1}$.
Since $X_1,...,X_k$ are independent, we know that $\mathbb{P}_{X_1}\otimes...\otimes \mathbb{P}_{X_k} = \mathbb{P} \circ (X_1,...,X_k)^{-1} $ $ (*)$. First, since $h$ is measurable we know that $h(X_1,...,X_n)$ is again a random variable.
How to show that $\int_{\Omega} h(X_1(\omega),...,X_k(\omega)) d\mathbb{P}(\omega) = \int_{\mathbb{R}^k} h(x_1,...,x_k) d(\mathbb{P}_{X_1}\otimes...\otimes \mathbb{P}_{X_k})(x_1,...,x_k)$? (everything is well defined since $h$ is bounded)
The statement is clear for indicator functions $h$ since we can just use $(*)$, and this means also that the statement is true for elementary functions. I'm stuck in trying to generalize the statement to positive measurable functions.