Independence of Random Variables 
*

*If $X$ and $Y$ are independent random variables so are also the random variables $f(X)$ and $g(Y)$ for $f$ and $g$ measurable and bounded functions.
The independence of $X$ and $Y$ implies:
$\Bbb{E}(f(X)g(Y)) = \Bbb{E}(f(X))\Bbb{E}(g(Y))$.
I don't see how the independence of $f(X)$ and $g(Y)$ follows from there.

*I also would like to see why the complements of $X$ and $Y$ will also be independent.
Thanks for any comment.
 A: $\newcommand{\E}{\operatorname{E}}$The independence of $f(X)$ and $g(Y)$ does not follow from the fact that $\E(f(X)g(Y))$ $= \E(f(X))\E(g(Y))$.  That last fact would imply that $f(X)$ and $g(Y)$ are uncorrelated, provided that $f(X)$ and $g(Y)$ have finite variances.  So you have
$$
\text{independence of }f(X)\text{ and }g(Y)\text{ entails } \E(f(X)g(Y))= \E(f(X))\E(g(Y))
$$
and
$$
\E(f(X)g(Y))= \E(f(X))\E(g(Y)) \text{ does not entail independence of }f(X)\text{ and }g(Y).
$$
The way in which $f(X)$ and $g(Y)$ are proved to be independent is this.  One must show that the events
$$
[f(X)\in A],\qquad [g(Y)\in B] \tag 1
$$
are independent for every pair of measurable sets $A$ and $B$.  That is what independence of random variables is.  Now recall that
$f^{-1}[A] = \{x\in\mathbb R: f(x)\in A\}$ and $g^{-1}[B] = \{x\in\mathbb R: g(x)\in B\}$.  (It doesn't matter here that $f$ and $g$ may fail to be one-to-one so that no inverse functions $f^{-1}$ and $g^{-1}$ exist; this is simply a definition of what the notation $f^{-1}[A]$ means.  Observe that $f(X)\in A$ if and only if $X\in f^{-1}[A]$, and similarly for $g$.  And the events
$$
X\in f^{-1}[A],\quad Y\in g^{-1}[B] \tag 2
$$
are independent because independence of the two reandom variables implies that.  Now observe that the two events in $(2)$, reported here to be independent, are the same events that appear in $(1)$.  Therefore the events in $(1)$ are independent.  And this holds for EVERY pair of sets $A$ and $B$.  Hence $f(X)$ and $g(Y)$ are independent.
As for complements: I know what the complement of an event is; I don't know what the complement of a random variable is.
