Is it true that ℙ{ F(X) ≤ u } ≤ u for all u in [0,1]?

Question

If X is an arbitrary (not necessary continuous) r.v. and F is its cdf, is it true that $\mathbb{P}(\{F(X)\leq u\})\leq u$ for all $u$ in $[0,1]$?

This feels like it should be straightforward but it has me confused

Answer 1

Always true. By monotonicity of $F$, $E\equiv F^{-1}[0,u]$ is either of the type $(-\infty, t)$ or of the type $(-\infty, t]$. Note that $t-\frac 1n $ is always in this interval so $F(t-\frac 1 n) \le u$ for all $n$. In the limit this gives $P(X<t)\le u$ or $P(X \in E)\le u$ when $E=(-\infty, t)$. Thus, $P(F(X) \le u)\le u$ in the first case. In the second just use the fact that $t \in (-\infty, t]=E$.