If Y is positive then show that $\{Y=+\infty\} \subset \{E[Y\mid \mathcal{G}]=+\infty\}$ where $Y$ ,lives on $(\Omega,\mathcal{F},P)$ and $\mathcal{G}\subseteq \mathcal{F}$.()
My attempt In order to show $A \subset B$. Its enough to show $B^c \subseteq A^c$ i.e $$ \{E[Y\mid \mathcal{G}]<+\infty\} \subseteq \{Y<+\infty\} $$
I set $A=\{E[Y\mid \mathcal{G}]<+\infty\}$ and clearly $A$ is $\mathcal{G}$ and hence $\mathcal{F}$-measurable .
If $A$ is not in $\{Y<+\infty\} $ almost surely then $A \cap \{Y=+\infty\}$ has a measure stricly greater than zero(assuming both $A$ and $\{Y=+\infty\}$ both have a strictly positive measure)
But on the set $\{Y=\infty\}$ we must have $E[Y\mid \mathcal{G}]=\infty$ i.e $1_{\{Y=\infty\}}E[Y\mid \mathcal{G}]=+\infty$ a.s.
In other words the set $A \cap\{Y=+\infty \}$ is null
And hence $P(A \cap \{Y=+\infty\})=0$
Is this proof correct?