Proof of a conjecture on the supports of Wigner distributions

In this note we prove that the Wigner distribution of an f is an element of L-2 (R-n) cannot be supported by a set of finite measure in R-2n unless f = 0. We prove a corresponding statement for cross-ambiguity functions. As a strengthening of the conjecture we show that for an f is an element of L-2 (R-n) its Wigner distribution has a support of measure 0 or infinity in any half-space of R-2n.