Notation and absolute continuity for probability measures A Question on Notation: 
I've realized that different notations are used in probability theory when evaluating an integral, and I am unsure as to how they "work" together, whether they're completely equiavalent or whether they do have subtle difference, and in which situations they should be used. 
Let $(\Omega, \mathcal{F}, P)$ be a probability space and $X$ a real random variable that describes the distribution with pdf $f$.
We are told that 
$\mathbb E[X]=\int_{\mathbb R}xf(x)dx$
and 
$\mathbb E[X]=\int_{\Omega}XdP$
but that 
$\int_{\Omega}XdP=\int_{\mathbb R}xf(x)dx$ only if $P_{X}$ is absolute continuous w.r.t $P$
Can anyone elaborate on this further. It still does not sit well with me. 
 A: The definition of the expectation of random variable $X$ is (if it exists):$$\mathbb EX=\int X\;dP\tag1$$so is an integral on the original probability space.
A random variable $X$ induces a probability measure on $(\mathbb R,\mathcal B)$ where $\mathcal B$ denotes the $\sigma$-algebra of Borel subsets of $\mathbb R$.
This probability measure is often denoted as $P_X$ and it is prescribed by:$$B\mapsto P(\{X\in B\})$$
A new probability space is introduced now: $(\mathbb R,\mathcal B,P_X)$.
Further there is a theorem that says that:$$\int X\;dP=\int x\;dP_X\tag2$$
For a probability measure $\mu$ on $(\mathbb R,\mathcal B)$ there might exist a non-negative Borel-measurable function $f:\mathbb R\to\mathbb R$ that satisfies:$$\int_Bf(x)dx=\mu(B)$$
If this is the case then it can be proved that:$$\int g(x)f(x)\;dx=\int g(x)\;d\mu\tag3$$for suitable functions $g$.
If for $P_X$ such a function exists then the function serves a so-called PDF (probability density function) and is mostly denoted as $f_X$. 
Application of $(2),(3)$ on the identity function then results in:$$\int g(x)f_X(x)\;dx=\int g(x)\;dP_X=\int g(X)\;dP$$
Doing this for the identity function gives:$$\int xf_X(x)\;dx=\int x\;dP_X=\int X\;dP=\mathbb EX$$
