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