# Can a variable affected by a random variable be also considered a random variable by extension?

I'm currently learning about instrumental variables analysis. One of its assumptions – the exclusion restriction – states that the instrument affects the outcome only through the exposure variable, and that the instrument is a random occurrence in nature. It basically emulates a randomized controlled trial wherein the instrument determines the exposure.

If it were an actual RCT, then treatment assignment is random. But since in instrumental variables analysis, the instrument is random and exogenous (supposedly) and that it determines the exposure, would exposure in such context be considered 'random' as well?

The definition of a random variable is that it is a measurable function from a probability space $$(\Omega, \mathcal F, p)$$ to $$\mathbb R$$. So if $$f:\Omega\to\mathbb R$$ is a random variable and $$g:\mathbb R\to\mathbb R$$ is the variable we are considering, then the composition $$g\circ f$$ is again a measurable function $$\Omega\to\mathbb R$$. So $$g$$ is considered a random variable in the sense that its composition $$g\circ f$$ with $$f$$ is a random variable.