Htykuut

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) $E[X]=int_Bbb R xf(x)dx=int_Bbb R xf(x)u(dx)=int_Bbb R xfdu=int_Bbb R xfrac{dP_X}{du}du=int_Bbb RxdP_x$

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 $f$.