The following question is from the exercises in a course on Stochastic Differential equations. We use "Stochastic Differential Equations" by Oksendal as a textbook.
An experiment consists of choosing a point in the following set.
$Ω=\{(x,y):x>0,x^2+y^2<1\}$
Let the $σ$-algebra F be the least σ-algebra such that all sets $\{(x,y)∈Ω:a≤x≤b \}$ and $\{(x,y)∈Ω:a≤y≤b\}$ are events, for any real a,b. Let P(A) of any $A∈F$ be proportional to its area. Define random variables $X(x,y)=x$ and $Y(x,y)=y$.
The question is:
State the conditional mean $Z=E\{X|Y\}$. Give an example of a Y-measurable event H with $0<P(H)<1$ and verify that the defining property of the conditional expectation holds, i.e. $E\{X⋅1H\}=E\{Z⋅1H\}$.
Now finding the conditional expectation I think I have a handle on, it is just about considering the geometry of $\Omega$. I can also state a $Y$-measurable event, but how would I show the defining property of the conditional expectation holds?
Thanks for looking!