Math Help - probability measure

1. 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?

2. Re: probability measure

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