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Suppose that $X_1,\ldots,X_n$ are independent random variables defined in some probability space $(\Omega,\mathcal F,P)$. Suppose that $f:\mathbb{R}^n\to\mathbb R$ is Borel measurable.

I think that is true that $$ \bigotimes_{i=1}^n (P\circ X_i^{-1})(f^{-1}(A))=P(f(X_1,\ldots,X_n)\in A), $$ for all $A\in\mathcal{B}(\mathbb R)$. Can you give a hint to prove that?

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The independence tells us that:$$P\circ\left(X_{1},\dots,X_{n}\right)^{-1}=\bigotimes_{i=1}^{n}P\circ X_{i}^{-1}$$

Then:

$\begin{aligned}\left(\bigotimes_{i=1}^{n}P\circ X_{i}^{-1}\right)\left(f^{-1}\left(A\right)\right) & =\left(P\circ\left(X_{1},\dots,X_{n}\right)^{-1}\right)\left(f^{-1}\left(A\right)\right)\\ & =\left(P\circ\left(X_{1},\dots,X_{n}\right)^{-1}\circ f^{-1}\right)\left(A\right)\\ & =\left(P\circ\left(f\left(X_{1},\dots,X_{n}\right)\right)^{-1}\right)\left(A\right)\\ & =P\left(f\left(X_{1},\dots,X_{n}\right)\in A\right) \end{aligned} $

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