Let $F:\mathbb R \to [0,1]$ be a distribution function of a probability measure $P$. This is,

$$F(x)=P((-\infty,x])$$

Then show that there is a random variable $X$, with

$$X : ((0,1],\mathcal B,\lambda)\to \mathbb R$$

(where $\mathcal B$ is the borel $\sigma $-algebra and $\lambda$ is Lebesgue measure), such that $P_{X}=P$

Let $F:\mathbb R \to [0,1]$ be a distribution function of a probability measure $P$. This is,

$$F(x)=P((-\infty,x])$$

Then show that there is a random variable $X$, with

$$X : ((0,1],\mathcal B,\lambda)\to \mathbb R$$

(where $\mathcal B$ is the borel $\sigma $-algebra and $\lambda$ is Lebesgue measure), such that $P_{X}=P.$

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 $P_{X}=P$

Let $F:\mathbb R \to [0,1]$ be a distribution function of a probability measure $P$. This is,

$$F(x)=P((-\infty,x])$$

Then show that there is a random variable $X$, with

$$X : ((0,1],\mathcal B,\lambda)\to \mathbb R$$

(where $\mathcal B$ is the borel $\sigma $-algebra and $\lambda$ is Lebesgue measure), such that $P_{X}=P$