I am interested in the correspondence between probability and measure theory. Currently, I know that a random variable is a measurable function, a probability function is a measure, etc.,
But, I am confused about whether the cumulative distribution function is also a measure like a probability function, or does it have any more accurate counterpart in measure theory?
CDFs are in one to one correspondence with probability measures in the sense that if you have a right continuous increasing function $F : \mathbb{R} \to [0, 1]$ with $\lim_{x \to -\infty}F(x) = 0$ and $\lim_{x \to \infty}F(x) = 1$, then $F$ is the cdf of a unique probability measure on $\mathbb{R}$. One method of proof is to let $U$ be uniform on $(0, 1)$, and let $X = F^{-1}(U)$, where $F^{-1} : (0, 1) \to \mathbb{R}$ is defined by $$F^{-1}(u) = \inf\{x \in \mathbb{R} : F(x) = u\}.$$ Then $X$ has CDF $F$.