Hello,

Sorry if I explain it in my own words, I don't know if you've studied it the same way... And maybe there are small typos, but not very important.

1) We know that if Z is

-measurable, then for any rv X (in L^2 I think), we have

So since Y is obviously

-measurable, E[E[XY|Y]]=E[YE[X|Y]] (*)

But there's something that says :

Let

be a

-algebra. For any

-measurable Z (positive), E[ZX]=E[Z E[X|Y]], and where X is positive. (this comes from the fact that E[X|Y] is the orthogonal projection of X over

, but you don't really need to know it if you haven't learnt this...)

So (*)=E[YX]

Exact same reasoning, under the condition that h is

-measurable.

I hope this is clear enough

*

is the smallest sigma-algebra that makes B measurable

** Note : a rv A is

-measurable iff there exists

which is

-measurable such that