How can I show that for every $\alpha \in [0,1]$ there is a $ B_{\alpha} \in \cal B(\mathbb R)$ with $\mu(B_{\alpha})=\alpha$?

Question

Let $\mu$ be a probability measure on $\mathbb R $ and $\mu(\{x\})=0$ for all $x \in \mathbb R$. How can I show that for every $\alpha \in [0,1] $ there is a $ B_{\alpha} \in \cal B(\mathbb R)$ with $\mu(B_{\alpha})=\alpha$?

Answer 1

IF the measure of all singletons is zero, the distribution function is continuous. You have zero covered since every singleton is of measure zero and you have 1 covered since the line minus a singleton has measure 1. Now invoke the Intermediate value theorem for the rest.