I recently learned the mathematical definition of a random variable, namely:

A random variable is a measurable function $X: \Omega \rightarrow \mathbb{R}$ whose domain $\Omega$ is equipped with a $\sigma-$algebra $\mathcal{F}$ and a probability measure $P$. 

This definition seems unintuitive to me, so I wanted to see how some of the famous classes of random variables are defined using the definition (e.g. Gaussian, Uniform). However, from what I understand in these cases there is no clear sample space $\Omega$ but rather a pdf which is used to define the function. That got me thinking why can't we just define a random variable as a probability measure on the Borel sets instead of using another measure on another $\sigma-$algebra and then requiring the function to be measurable?