When I looking expectation value in wikipedia
I find that it has several expression for expectation value
(1).$E[X]=\int_\Omega X(\omega)dP(\omega)$
(2).$E[X]=\int_\Bbb R xf(x)dx$
and a random variable is a function that transform a probability space to another space is built up on real number
i.e ($\Omega,\mathcal F,P)\to(\Bbb R,\mathcal B,P_X$)
so,it seems (1) is a expression based on the probability space $(\Omega,\mathcal F,P)$
(2) is for probability space $ (\Bbb R,\mathcal B,P_X$)
and I know $f=\frac{dP_X}{du} $ is a Radon–Nikodym derivative where $u$ is Lebesgue measure
Now I am trying to rewrite the expression from (2) to (1)
(3) $E[X]=\int_\Bbb R xf(x)dx=\int_\Bbb R xf(x)u(dx)=\int_\Bbb R xfdu=\int_\Bbb R x\frac{dP_X}{du}du=\int_\Bbb RxdP_x$
Is that (3) correct?
and how do I rewrite the right hand side of (3) to (1)?
$P_X$ is defined by $P_X(B)=P(X^{-1}(B))$.Ffrom this we can show that $\int xdP_X=\int XdP$ (by simple function approximation). It should be noted that 1) is more general than 2) in the sense it does not require existence of $f$.