Let be a probabilty space and be a random variable with distribution . Show that is a probabilty measure on . That is, verify the three axioms of probabilty for , using the collection of "events" on .
I've become stuck with the question. My teacher said that I need to use the fact that is a probability measure and that is a random variable. I've racked my brain but can't work it out Whats more annoying is that the answer is staring me right in the face.
I know that the 3 axioms of probability are
(1) for some event A
(2)
(3) For any sequence which are disjoint
I just can't get my head around this and any help would be greatly appreciated.
I'm just learning about probability and measure theory myself, and I am not a mathematician. So here's my best shot at it:
We have that , and I suppose that makes an extended random variable. Now, induces a probability measure on , namely . The distribution function of is then . Now, because for , , the distribution corresponds to the measure . We should have as and as . Is that of any help?
By the way, which book do you use to study measure theory?