[proofplan]
Test independence on arbitrary measurable target sets and pull them back through the measurable maps $f_i$.
[/proofplan]
[step:Pull back target events]
Let $H_i\in\mathcal H_i$ for $1\le i\le n$. Since $f_i$ is measurable, $f_i^{-1}(H_i)\in\mathcal E_i$. Also
\begin{align*}
\{f_i(X_i)\in H_i\}=\{X_i\in f_i^{-1}(H_i)\}.
\end{align*}
[/step]
[step:Apply independence of the original variables]
Using the independence of $X_1,\ldots,X_n$,
\begin{align*}
\mathbb P\left(\bigcap_{i=1}^n\{f_i(X_i)\in H_i\}\right)
&=\mathbb P\left(\bigcap_{i=1}^n\{X_i\in f_i^{-1}(H_i)\}\right) \\
&=\prod_{i=1}^n\mathbb P(X_i\in f_i^{-1}(H_i)) \\
&=\prod_{i=1}^n\mathbb P(f_i(X_i)\in H_i).
\end{align*}
This is the definition of independence for $f_1(X_1),\ldots,f_n(X_n)$.
[/step]
