Let be an arbitrary space, and let , be two probability measures on . Suppose there is a common set on which .
Now suppose there are such that . Can anything be said of and , or and ?
My initial intuition was that the answer should be yes since and both have full and full measure respectively, and so there should be "enough" overlap for to have at least positive measure. But I can't seem to prove this - so I'm interested in whether anything can be said of these quantities (either alone or possibly together, e.g. is it possible for ) and whether there is any clever counterexample for which and can be any arbitrary number in [0,1].