Random variable function is random variable?

Question

I'm starting to study random variables and during class my teacher mentioned that a non-descending random variable function is a random variable. Then he said that $|X|$ and the indicator function $I(A_x)$ would be random variables. Intuitively I can see they are. But how do I prove it in the general case, or for these two examples?

Answer 1

As random variable (let’s consider one-dimensional case for simplicity) defined to be a measurable function from sample space $\Omega$ to $\mathbb{R}$ with corresponding $\sigma$-algebras (usually, it is Borel $\sigma$-algebra for $\mathbb{R}$).

It’s known that composition of two measurable functions is again measurable, so for random variable $X$, for any measurable function $f$ from $\mathbb{R}$ to $\mathbb{R}$ (again, with respect to Borel $\sigma$-algebra) $f(X) : \Omega \to \mathbb{R}$ is also a random variable. In particular, all continuous $\mathbb{R} \to \mathbb{R}$ functions are measurable.