Intuitively I expect this to follow from countable additivity, but there are ideas I can't rule out such as:
- Select a real number r from the uniform distribution over [0, 1]. If r is exactly 0.5, then let q = 0, else let q = 1
- Select r as above. If r is rational let q = r, else let q = 0
EDIT: @GEdgar writes, "In both of your examples, q has discrete distribution. In example 1, q=1 with probability one. In example 2, q=0 with probability one." My thought was that a random variable that is "almost surely" zero might not be the same as one that is identically zero. But the truth is I have no idea what sort of tortured distribution might work here, much like I never would have come up with the Cantor distribution if someone had asked me whether all random real variables have to be discrete or continuous.