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

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SUMMARY

The discussion focuses on proving that if a random variable Y is F0-measurable, then the conditional expectation E[Y|F0] equals Y, given that E[|Y|] is finite. The definition of conditional expectation is provided, emphasizing that E[Y|F0] is an F0-measurable random variable satisfying the equation E[IFY] = E[IFE[Y|F0]] for all F in F0. Participants suggest that the inquiry resembles an exercise from a measure theoretic probability course, indicating a need for foundational understanding or textbook reference.

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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([tex]\Omega[/tex],F,P) and satisfying E[|Y|]<∞. Let F0 be a sub-[tex]\sigma[/tex]-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[IFY]=E[IFE[Y|F0]] for all F [tex]\in[/tex] F0

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

I appreciate any response.
 
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anybody??
 
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.
 

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