A random variable is a function X(w) taking (eg) real values. In your probability space you already have an ambient measure space and an ambient probability measure P which takes sets in the measure space to [0,1]. The pdf is then the function defined on sets P(invX(q)). invX is a set valued inverse.
Ok, consider coin flips. Then X takes each element of sample space either to 1 or -1. Set values inverse of 1 is the sets that map to 1. Then we get the ambient probability measure of them.
You don’t really have to cope with measure theory in full to take this tiny step.
A random variable is a function X(w) taking (eg) real values. In your probability space you already have an ambient measure space and an ambient probability measure P which takes sets in the measure space to [0,1]. The pdf is then the function defined on sets P(invX(q)). invX is a set valued inverse.
Ok, consider coin flips. Then X takes each element of sample space either to 1 or -1. Set values inverse of 1 is the sets that map to 1. Then we get the ambient probability measure of them.
You don’t really have to cope with measure theory in full to take this tiny step.