  The integral of the Wigner function over a subregion of the phase-space of a
quantum system may be less than zero or greater than one. It is shown that for
systems with one degree of freedom, the problem of determining the best
possible upper and lower bounds on such an integral, over all possible states,
reduces to the problem of finding the greatest and least eigenvalues of an
hermitian operator corresponding to the subregion. The problem is solved
exactly in the case of an arbitrary elliptical region. These bounds provide
checks on experimentally measured quasiprobability distributions.
