# Borel $\sigma$-algebra, generating sets and random variables

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.

1. First form a $$\sigma$$-algebra on $$[0,1]$$, call it $$\sigma_Y$$
2. 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 $$B$$.