  It is well known that the independent random variables $X$ and $Y$ are
uncorrelated in the sense $E[XY]=E[X]\cdot E[Y]$ and that the implication may
be reversed in very specific cases only. This paper proves that under general
assumptions the conditional uncorrelation of random variables, where the
conditioning takes place over the suitable class of test sets, is equivalent to
the independence. It is also shown that the mutual independence of
$X_1,\dots,X_n$ is equivalent to the fact that any conditional correlation
matrix equals to the identity matrix.
