Please explain in details, and I don't want answers copied from other WEBSITES ANSWERS. Thanks!

#1

Let (1LT, P) be a probability space. Let X be a random variable in this probability space, and g, C 9-" be a sub-o-algebra. Show that X is g:- measurable if and only if {700 C 9. Show that the intersection of two o-algebras is a o-algebra; that is, show that if F and g are two o-algebras then their intersection F ng is also a o-algebra