So my book defines that a random variable $X$ has absolutely continuous distribution $F$, if it can be represented as $F(x)=\int_{-\infty}^x p(\mu)\,d\mu$ for some non-negative and integrable function $p$.
I was wondering if this definition is equivalent to asking for the pushforwards $P\circ X^{-1}$ to be absolutely continuous with respect to the Lebesgue measure.
Im kind of convinced that it has to be the Lebesgue measure since it completes the Borel measure but I'm not sure.