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 $X$.