Let $\mu$ be a probability distribution on $(\mathbb R, \mathcal B(\mathbb R))$ and $X$ be a random variable with distribution $\mu$. According to the theory developed in Section 3.5 of Folland's Real Analysis, $\mu$ is absolutely continuous w.r.t. Lebesgue measure iff the cdf $F_X$ is absolutely continuous.
Assume that $F_X$ is known. What are tractable conditions on $F_X$ which guarantee that $F_X$ is absolutely continuous ?
For example, I know that the following condition is sufficient (since it implies that $F_X$ is Lipschitz):
- $F_X$ is continuous over $\mathbb R$, $F_X$ is differentiable except perhaps at finitely many points, and $(F_X)'$ is bounded.
Are the following conditions also sufficient ?
- $F_X$ is continuous over $\mathbb R$ and $F_X$ is differentiable except perhaps at finitely many points.
- $F_X$ is continuous over $\mathbb R$ and $F_X$ is differentiable except perhaps at countably many points.
- $F_X$ is continuous over $\mathbb R$ and $\int_{\mathbb R} F_X'(x) dx = 1$.
Note that $F_X$ already has some regularity (non-decreasing, cadlag, limits at $\pm \infty$, differentiable a.e.).
Best Answer
It is well known (e.g. as a consequence of Lebesgue's decomposition theorem) that any continuous cdf can be decomposed as $F=F_a+F_s$, where $F_a$ is absolutely continuous and $F_s$ has derivative $0$ almost everywhere. This implies that 2–4 are sufficient.