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Please help me prove $E(X|Y)=0$ given that, for any measurable function $g$: $$E[Xg(Y)]=0$$

I have been trying using a definition of conditional expectation, but it does not seem to work.

Thanks!

mookid
  • 28,236
bankrip
  • 566

1 Answers1

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Hint: there is a measurable function $h$ such as $E[X|Y] = h(Y)$. And you also have, for any well behaved $g$, $$ E[Xg(Y)] = E[E[X|Y] g(Y)] $$

mookid
  • 28,236
  • I know that $E[Xg(Y)]=E[E[X|Y]g(Y)]$ is from the definition of $E(X|Y)$. It, then, implies that $E[E[X|Y]g(Y)]=0$ for any well behave $g$, but how does this implies $E(X|Y)=0$? – bankrip Nov 18 '14 at 21:35