Skorokhod Representation of a Random Variable

See 3.12 of Williams.

Let $X^{-}(\omega)$ $= \inf \{ x \mid F(z) \geq \omega \}$ where $F : \mathbb{R}$ $\rightarrow [0,1]$ and

1. $x \leq y \implies F(x) \leq F(y)$.
2. $\lim_{x \rightarrow \infty} F(x)= 1$ and $\lim_{x \rightarrow -\infty} F(x) = 0$.
3. $F$ is right continuous.

Then by the proof $X^{-}(\omega) \leq c \implies \omega \leq F(c)$. Thus $c \not \in \{ y \mid F(y) < \omega \}$ and is therefore an upper bound (if not then $\exists y$ such that $F(y) < \omega$ and $c < y$ but by monotonicity $F(c) \leq F(y)$). Therefore $c \geq \sup \{ y \mid F(y) < \omega \}$.

On the other hand suppose $c < \inf \{ x \mid F(x) \geq \omega \}$ then $F(c) < \omega$ (if not then $F(c) \geq \omega$ but then $\inf \{ x \mid F(x) \geq \omega \}$ is a lower bound for all such $c$ which would imply $\inf \{ x \mid F(x) \geq \omega \} \leq c$). Now $\sup \{ y \mid F(y) < \omega \}$ is an upper bound for any $x$ such that $F(c) < \omega$ so $c \leq \sup \{ y \mid F(y) < \omega \}$. Now suppose $c = \sup \{ y \mid F(y) < \omega \}$ then $F(c + 1/n) \geq \omega$ and by right continuity this implies $F(c) \geq \omega$. Thus we must have $c < \sup \{ y \mid F(y) < \omega \}$