Understanding the definition of a measurable function

Question

I’m having trouble understanding the definition of a probability measurable function. The definition says that the preimage of events in the sigma algebra on the range must be an event in the sigma algebra on the domain. Why is this definition significant? What’s it trying to say?

Answer 1

Assume $X$ is a probability space, and $f:\>X\to{\mathbb R}$ is some scalar function. For a given number $a\in{\mathbb R}$ we want to be able to talk about the probability $P\bigl[f(x)\leq a\bigr]$. This probability is equal to the measure of the set $f^{-1}({\mathbb R}_{\leq a})\subset X$:$$P\bigl[f(x)\leq a\bigr]=\mu\left( f^{-1}\bigl({\mathbb R}_{\leq a})\right)\ ,$$ and similarly $$P\bigl[a\leq f(x)\leq b]=\mu\left( f^{-1}([a,b])\right)\ .$$ Therefore we need the property that $f^{-1}$ of measurable sets in ${\mathbb R}$ is measurable in $X$.