## Prove {X>0} implies {E(X|G)>0}

Let $\displaystyle X \geq 0$ be a random variable on the probability space $\displaystyle (\Omega, F, \mathbb{P})$ and let $\displaystyle G \subset F$ be a sub-$\displaystyle \sigma$-algebra. Show that $\displaystyle \{X>0\} \subset \{E(X|G) > 0\}$, up to an event of zero probability. Show that $\displaystyle \{E(X|G) > 0\}$ is actually the largest G-measurable event that contains the event $\displaystyle \{X>0\}$, up to zero probability events.