Let be a probability space, and let and be two -algbras on . Suppose and . The -algbras and are independent if for any , any , . Suppose and are independent -algbras, and a r.v. is measurable from both to and from to (where is the set of all real numbers and is the Borel -algebra on ). Show that is almost surely a constant. i.e. for some constant c.
Anyone can help? Thanks in advance.