Let be a probability space, let be an -random variable, let be a sub-sigma algebra of , let , let be the intersection of all the sets in that contain , and suppose that . Then is it true that ?
I just wrote this statement down based on my intuition of what conditional expectation means, and I want to verify if it’s actually true. Note that sigma algebras need not be closed under uncountable intersections, so need not be an element of . But I assumed is at least an element of so that is meaningful.