Absolute continuity with respect to Lebesgue Measure of discrete random variable?

Question

Show that the distribution function of any discrete random variable AC with respect to the counting measure but not AC with respect to the Lebesgue measure! Show that any distribution function that has discontinuity points cannot be AC with respect to the Lebesgue measure

Answer 1

Hint for Part 2: Just follow the $\epsilon - \delta$ definition of absolute continuity. Suppose $F$ has a jump at point $x_0$, i.e. $F(x_0) - F(x_0-) = \eta > 0$ and $F$ is absolutely continuous. Then fix $\epsilon > \eta$. As $F$ is absolutely continuous there exists $\delta > 0$ such that whenever $\sum_{i=1}^k (x_i - y_i) \le \delta$ for any collection $\{x_1, y_1\}, \cdots, \{x_k,y_k\}$, then $\sum_{i=1}^k |F(x_i) - F(y_i)| \le \epsilon$. Now show that this leads to a contradiction, if you choose points near the jump (e.g. $\{x_0, x_0+\delta\}$).

This will also help you to show why distribution function of discrete random variable is not absolutely continuous w.r.t Lebesgue measure.