I am reading a book (Hilbert Space Methods in Probability and Statistical Inference by Small) which says that random variables can be viewed as functions in the hilbert space $L^2$ with the inner product defined as expectation $\left<X,Y\right> = E\left(XY\right)$. However, inner products must be finite as far as I know. Does this mean that the hilbert space view of random variables is unable to handle cauchy random variables and others with non-finite moments?
(this question moved from stats.stackexchange)