How does a probability density function of a random variable is induced due to Radon-Nikodyn derivative?

Question

If someone can please clarify the following for me: Given a probability space $(\omega,\Sigma,P)$ and a continous random variable $X:w \rightarrow \mathcal{R}$. I have read the statement of the Radon-Nikodyn Theorem, but I still can't see how does one relate it to the existance of a probability density function. Can someone please elaborate on this? Thanks

Answer 1

Let $\mathbb P_X$ the measure on $\mathbb R$ defined by $$\mathbb P_X(B):=\mathbb P\{X\in B\},$$ for all Borel set $B\subset \mathbb R$.

Definition Let $m$ denote the Lebesgue measure on $\mathbb R$. Then, $X$ is continuous if $\mathbb P_X$ is absolute continuous with respect to $m$.

Therefore, by Radon-Nikodym, there is $f:\mathbb R\to \mathbb R$ s.t. $$\frac{\mathrm d \mathbb P_X}{\mathrm d m}=f,$$

i.e. $$\mathbb P_X(B)=\int_B f(x)m(\mathrm d x).$$ Therefore, $X$ has a density $f$.