Please note that I'm working with the following definition of random variable, which allows for a codomain other than $\mathbb{R}$.
Definition: Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $(\Omega', \mathcal{F}')$ a measurable space. Then a random variable is a $\mathcal{F}/\mathcal{F}'$-measurable mapping $f: \Omega \to \Omega'$.
I know that for $(\Omega', \mathcal{F}') = (\mathbb{R}, \mathcal{B}(\mathbb{R}))$, the sum of two random variables on $(\Omega, \mathcal{F}, \mathbb{P})$ is also a random variable on the same probability space. Indeed, there are already proofs of this theorem here on Math.SE. But that leaves the following question open.
Question:
- Is it generally true that the sum of two random variables is a random
variable? - If not, what's a simple counterexample?
Best Answer
Counterexample:
Let $\Omega'$ be some set equipped with $\sigma$-algebra $\{\varnothing,\Omega'\}$ and let it be that there is no further structure present on $\Omega'$.
Then automatically every function $f:\Omega\to\Omega'$ is measurable but $f+f$ is not even defined.