How to Prove E[Y|F0]=Y When Y is F0-Measurable?

[umutk]

I need help about conditional expectation for my research. I get stucked on this point. Could anyone show me how to prove that:
"Let E[|Y|]<∞. By checking that Definition is satisfied, show that if Y is measurable F₀, then E[Y|F₀]=Y."

Def: Let Y be a random variable defined on an underlying probability space($$\Omega$$,F,P) and satisfying E[|Y|]<∞. Let F₀ be a sub-$$\sigma$$-algebra of F. The conditional expected value of Y given F₀,denoted E[Y|F₀],is an F₀-measurable random variable that also satisfies:E[I_FY]=E[I_FE[Y|F₀]] for all F $$\in$$ F₀

Note that: Red Fs are sets, but black Fs are sigma-algebras.

I appreciate any response.

 

[umutk]

anybody??

 

[Pere Callahan]

Welcome to PF.
Are you really trying to tell us that is research? To me it sounds like an exercise from a measure theoretic probability course.
I suggest you show what you have tried so far or look up the answer in a textbook.