Definition of random variable (measure theoretic)

Question

My book defines a random variable $X$ as a real measurable function on a probability measure space $(\Omega, F, P)$. My question is as follows: is the probability measure $P$ defined over the elements of $F$, or over the elements of $X(F)$? Or is there no difference?

Answer 1

The measure is defined in the measure space, that is, $P$ is a measure in $\Omega$, not in the codomain of $X$.

However the measure $P$ induces a Lebesgue-Stieltjes measure $\mu_X$ in the codomain of $X$ by

$$\mu_X(A):=P(X^{-1}(A))$$

for any Lebesgue measurable set $A\subset\Bbb R$. This induced measure $\mu_X$ defines the distribution of $X$ via $F_X(c):=\mu_X((-\infty, c])$, and so we have that

$$ P(X^{-1}(-\infty,c])=\int_{X^{-1}(-\infty,c]} dP=\int_{(-\infty,c]}d\mu_X $$

where we generally use the shorthand $P(X\le c):=P(X^{-1}(-\infty,c])=F_X(c)$ to refer to "the probability that $X$ would be equal or less than $c$" and $dF_X:=d\mu_X$ is a common notation for the Lebesgue-Stieltjes measure.

When $F_X$ haves a continuous derivative then $dF_X=F_X'(x)\,dx$ where $dx$ is the Lebesgue measure on $\Bbb R$ and $f_X:=F'_X$ is called the density of $X$.

I hope you get all clear now.

NOTE: observe that above we require that $X$ is a measurable function, that is, that $X^{-1}(A)\in F$ for any Lebesgue measurable set $A$.