I want to show$X : ((0,1],\mathcal B,\lambda)\to \mathbb R$ is random variable

Question

Let $F:\mathbb R \to [0,1]$ be a distribution function of a probability measure $P$ $(i.e.,F(x)=P((-\infty,x])) $. Then show that There is a random variable $X : ((0,1],\mathcal B,\lambda)\to \mathbb R$,(where $\mathcal B$ is the borel $\sigma $-algebra and $\lambda$ is Lebesgue measure) ,such that $P_{X}=P$

Answer 1

This is a standard result that you should be able to find in any book on measure-theoretic probability. Look in chapter 1, section 2 here. It is Theorem 1.2.2.