How to show that $\{E[Y\mid \mathcal H] = 0\} \subset \{Y = 0\}$ almost surely?

Question

I'm stuck on this exercise from Probability Essentials (Jacod and Protter).

I know that Y is positive random variable.

I tried to show that $P(\{E[ Y\mid \mathcal H] = 0\}) \le P(\{Y = 0\})$ but without success. I'm also asked to show that $\{E[Y\mid \mathcal H] = + \infty\} \supset \{Y = +\infty\}$ almost surely.

Answer 1

Let $A=\{E[Y|\mathcal H]=0\}$. Then $$E[Y1_A]=E[E[Y1_A|\mathcal H]]=E[1_AE[Y|\mathcal H]]=0.$$ So $Y1_A=0$ a.s. since $Y\geq 0$, which means $Y=0$ a.s on $A$ so $P(A)\leq P(Y=0).$

For the other part, let $B=\{Y=+\infty\}$. By the definition of conditional expectation, for all $C\subset B$ which lies in the $\sigma-$field, we have $$\int_CE[Y|\mathcal H]\,dP=\int_C Y\,dP=+\infty.$$ Hence $E[Y|\mathcal H]=+\infty$ a.s. on $B=\{Y=+\infty\}$.