This comment was posted to reddit on Jan 02, 2015 at 10:45 pm and was deleted within 6 hour(s) and 10 minutes.

Measure-theoretic probability theory - Polya's urn

A bag contains red and green balls. Initially, there is one ball of each colour. A ball is drawn from the bag, its colour noted, and then it is returned to the bag together with a new ball of the same colour. Let Rn denote the number of red balls in the bag after n additions, and set Sn = Rn/(n + 2)

The homework question was prove that Sn is a martingale w.r.t. Xn which is the random variable corresponding to the ball drawn at time n. And also to prove that Sn converges a.s. to some random variable and what the distribution of the limit r.v. is.

But I'm not interested in that, although some of you might be, since I think I can figure those out (strange given my next question). My question is more basic:

My question is, what probability space are we working in when working with this problem? Is it the standard space (2N, F inf, P) when working with the infinite coin toss problem or are we working with a different sigma algebra or probability measure, because each term in the infinite sequence in 2N is not independent of the previous term?

Edit: To clarify, 2N is the set of all infinite 0-1 sequences. I think Finf can remain as the sigma-algebra used in the coin tossing problem, which is just the sigma algebra generated by the basic sets (if you want to know what I mean, ask me and I'll type it out, because it's quite complicated). The probability measure used in the coin tossing problem is simply one extended by Caratheodory's theorem from the function Po defined by Po(Aj1j2j3,...,jn) = 1/2n for any sequence j1 to jn and where Aj1,...,jn = {w in 2N such that w1 = j1, w2= j2 ,... , wn = jn}

I somehow managed to prove that Sn is a martingale with respect to the variable Xn of the ball drawn at the nth addition. But I think the way I found the probability P(Xn = 1) and P(Xn = 0) was very intuitive (I just said that the probability that the nth ball drawn is red is the ratio of red balls over the total balls in the urn) - I would like to know how we can define the probability measure P rigorously.