Simple question about independence and complex exponential function

Question

Let $X,Y$ be Independent random variables. Why does their Independence implie the Independence of $e^{iuX}$ and $e^{iuY}$? $u$ is real in this case.

Answer 1

The function $f:\mathbb{R}\to\mathbb{C}$ which is defined by $f(x)=e^{iux}$ is Borel measurable. Hence if $B_1,B_2\subseteq\mathbb{C}$ are Borel sets then $f^{-1}(B_1),f^{-1}(B_2)\subseteq\mathbb{R}$ are Borel, and we have:

$P(f(X)\in B_1,f(Y)\in B_2)=P(X\in f^{-1}(B_1),Y\in f^{-1}(B_2))=$ $=P(X\in f^{-1}(B_1))P(Y\in f^{-1}(B_2))=P(f(X)\in B_1)P(f(Y)\in B_2)$

We used the independence of $X$ and $Y$ here.