Clarification of Expectation

Let \(X\) be a random variable on the probability space \((\Omega, \mathcal{B}, P)\).  Recall that the distribution of X is the measure
\[F := P \circ X^{-1}\] on \((\mathbb{R},\mathcal{B}(\mathbb{R}))\) defined by
\[F(A)=P\circ X^{-1}(A) = P[X\in A].\]
The distribution function of \(X\) is
\[F(x):= F((-\infty,x])=P[X\leq x].\]  Note that the letter "F" is overloaded in two ways.

An application of the Transformation Theorem allows us to compute the abstract integral
\[E(X) = \int_\Omega XdP\] as
\[E(X) = \int_\mathbb{R} xF(dx),\] which is an integral on \(\mathbb{R}\).

More precisely,
\[E(X) = \int_\Omega X(\omega)P(d\omega)=\int_\mathbb{R} x F(dx).\]
Also given a measurable function \(g(X)\), The expectation of \(g(X)\) is
\[E(g(X)) = \int_\Omega g(X(\omega))P(d\omega)=\int_\mathbb{R} g(x) F(dx).\]
Instead of computing expectations on the abstract space \(\Omega\), one can always compute them on \(\mathbb{R}\) using \(F\), the distribution of \(X\).