Let $X:(\Omega,\mathcal {F})\to (\mathbb{R}^n,\mathcal{B}(\mathbb{R}^n))$ denote a random variable defined on some probability space $(\Omega, \mathcal{F},P)$.
By Lebesgue's theorem, the distribution of $X$ may be decomposed as $X(P)=f dm + \mu$ with $m$ representing the Lebesgue measure and $\mu \perp m$. Also, letting $F_X:\mathbb{R}^n\to [0,1]$ denote the CDF of $X$, we know that $F_X$ is almost everywhere differentiable.
Question
What is the relationship between $F_X'$ and $f$?
Whenever $n=1$, it holds that $F_X'=f$ almost everywhere (use, e.g., Theorem $3.22$ in Folland's Real Analysis to realize this), but I am curious about the case $n>1$.