On the conditional expectation E(X X + W) in the case of independent random variables X, W

Let X, W be independent real-valued random variables with finite expectation and E(W) = 0. We prove that only in the case W=0 the conditional expectation E(XX+W) coincides with X+W. The result is a consequence of the following cancellation theorem: Let P, Q, R be Borel probability measures on the real line such that the support of Q resp. R is contained in {x[less-than-or-equals, slant]0} resp. {x[greater-or-equal, slanted]0}; then P * Q = P * R implies that Q = R(=[delta]0).