Let us consider a typical real-valued random variable Y defined on a probability space (\Omega,\mathcal{F},\mathbb{P}). Let X be another such random variable, and assume X,Y\in L^{2}(\mathcal{F}). We emphasize the \sigma-algebra \mathcal{F} because this will be the structural element of our probability space subject to frequent change; in particular, the measure and sample space \Omega will remain constant (we ignore the issues that arise with this assumption when additional random variables are added to our universe, and implicitly assume \Omega and \mathbb{P} have been properly extended in a way that is consistent with all other previous random variables in play).
The space L^{2}(\mathcal{F}) is a Hilbert space with norm
(X,Y)=\mathbb{E}[XY]=\int_{\Omega}X(\omega)Y(\omega)\;d\mathbb{P}(\omega). The integral defining this inner product can be calculated as a Lebesgue integral on the range of the random vector (X,Y) (in particular, an integral over \mathbb{R}^{2}) in the usual way by changing variables and using the appropriate push-forward (distribution) measures of X and Y (and densities if the distributions are absolutely continuous with respect to Lebesgue measure).
