This is a exercise of Shiryaev's Probability on page 233:
Suppose that the random elements $(X, Y)$ are such that there is a regular distribution $P_x(B)=P(Y\in B\mid X=x)$. Show that if $E|g(X,Y)|<\infty$ then $E[g(X,Y)\mid X=x]=\int g(x,y)P_x(dy) \text{ ($P_x$-a.s.)}$
PS: I think there is a typo. Should we replace "$P_x$-a.s." by "$P_X$-a.s."?
I firstly tried to prove it for indicator function. For $g(x,y)=I_B(x,y)$, $B\in \mathcal{B}(R^2)$, I should prove for any $A\in \mathcal{B}(R)$, $\int_A E[I_B(X,Y)\mid X=x]P_X(dx)=\int_A\int I_B(x,y)P_x(dy)P_X(dx)$ Then, $LHS=\int_{\{X\in A\}}I_B(X,Y)dP$ For the RHS, let $B_x=\{y\mid (x,y)\in B\}$, I deduce that $\begin{align} RHS &=\int_A\int I_{B_x}(y)Px(dy)P_X(dx)\\ &=\int_A P_x(B_x)P_X(dx)\\ &=\int_A P(Y\in B_x\mid X=x)P_X(dx)\\ &=\int_A E(I_{B_x}(Y)\mid X=x)P_X(dx)\\ &=\int_A E(I_{B}(x,Y)\mid X=x)P_X(dx) \end{align}$ This problem is the origin of my another problem: "An equality about conditional expectation"
In that thread, did has pointed out that $E(I_{B}(x,Y)\mid X=x)$ is meaningless, I agree with him. But what's wrong with my deduction above?
By the way, in the thread mentioned above, StefanHansen have recommend me the book Probability With a View Towards Statistics, Volume II by J. Hoffmann-Jørgensen to study the concept of regular conditional probabilities/distributions, but I can't find a copy of it around me. Can anyone introduce me other materials which has an excellent treatment of regular conditional probabilities/distributions?
Thank you very much!