Prove that $B={\rm{E}[1_A \mid G)=0}$

Question

Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space, and $G\subset \mathcal F$ a subsigma algebra. For an $A\in \mathcal F$ consider the event:

$B=\{\rm{E}[1_A \mid G)=0\}$

Prove that almost surely $B\subset A^c$

Intuitively this is true because the definition of the sigmas algebras and conditional expectation, but im having problems on concrete the ideas. Thanks for the help.

Answer 1

The statement $B \subset A^{c}$ almost surely means that $P(A\cap B)=0$. To prove this note that $P(A\cap B)=E(1_A 1_B)=E(E(1_A 1_B|G))=E(1_BE(1_A|G))$ since $B$ is already measurable w.r.t. $G$. Hence $P(A\cap B)=E((1_B)(0))=0$