Some preliminary definitions:
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space.
Let $\mathcal B(\mathbb{R})$ be the Borel $\sigma$-algebra on $\mathbb{R}$.
I define $\mathcal B (\mathbb{R})$ to be the smallest $\sigma$-algebra generated by the collection of sets $\{ (-\infty, a] : a \in \mathbb{R} \}$. See optional info
$X : \Omega \mapsto \mathbb{R}$ is a random variable if and only if for every $B \in \mathcal B(\mathbb{R})$, $X^{-1}(B) = \{ \omega \in \Omega : X(\omega) \in B\} \in \mathcal{F}$ or in other words, $X^{-1}(B)$ is measurable w.r.t probability measure $\mathbb P$.
Now, assume $X$ is a random variable as per the previous definition. Hence, every $B \in \mathcal B(\mathbb{R})$ is a measurable set and define the measure $\mathbb{P}_X(B) = \mathbb{P}(X^{-1}(B))$.
Define $Y: \mathbb{R} \mapsto [0,1]$ by $Y = F_X(X)$, where $F_X$ is CDF of $X$.
Show that $Y$ is a random variable.
My thoughts about solution:
- First form a $\sigma$-algebra on $[0,1]$, call it $\sigma_Y$
- Then show that for every $C \in \sigma_Y$, $Y^{-1}(C) \in \mathcal{B}(\mathbb{R})$ or equivalently, $Y^{-1}(C)$ is measurable w.r.t probability measure $\mathbb{P}_X$
EDIT:
consider $\sigma_Y$ to be the $\sigma$-algebra generated by the collection $\{ (0, a] : a \in (0,1) \}$
Define $F_X^{-1}(a) = \sup\{x \in \mathbb R : F_X(x) \leq a \}$ for $0 < a < 1$.
For $(0, a]$ where $0 < a < 1$, we have $Y^{-1} \left( (0, a] \right) = \left(-\infty, F_X^{-1}(a) \right ] $ which is clearly $\in \mathcal{B}(\mathbb{R})$.
Is this complete proof?
$F_X$ is Borel measurable because it is increasing. [$F_X^{-1}(I)$ is an interval whenever $I$ is an interval]. Hence $(F_X(X))^{-1}(B)=X^{-1}(F_X^{-1}(B))$ is measurable for any Borel set $B$.