Let . Let . By the hypothesis, the set is non-empty. What about ?
Let P a probability measure and R a random variable on a certain space.
P(.) only takes on the values 0 and 1 for all events, I have to show that there is a number a, such that the event T=a, has probability one.
My idea was that P(.) can not be zero for all values, because the sum of P(.) must be 1. How do I work this out rigorously?