
probability measure
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?

Re: probability measure
Let $\displaystyle f:x\mapsto P(R\leq x)$. Let $\displaystyle S=\left\{x\in\mathbb R: f(x)=1\right\}$. By the hypothesis, the set $\displaystyle S$ is nonempty. What about $\displaystyle \inf S$?