Probability of an event in infinite trials needs to be formalized?

Question

Lets flip a coin an infinite number of times. Now we want to find out the probability that, for example, in the first n attempts it falls heads. I want to formalize this. So I define my measure space for one single flip as A={heads, tails}, The sigma algebra to be the power of this set, and p, my probability measure, as p(heads)=p(tails)=0.5. Now, if we want to work with infinite trials, I define the sample space as \product_{i=1}^\infty A. The sigma algebra in this case seems to be that generated by rectangles, each of which is the product of subsets off A and all except a finite number of them are equal to A(as in the product topology). My question is: how can we define a probability measure in this sigma algebra? Is there any other way to formally attack this problem? If there is, what is it? I think this the same problem as: Given a collection of measure spaces {A_i, B_i, p_i}, i=1,..., how to define the infinite product measure space of these?

Answer 1

You are along the right track. Let $\Omega$ be the set of infinite sequences $\{0,1\}^{\mathbb{N}}$. Let your $\sigma$-algebra be defined by taking the smallest $\sigma$-algebra containing all cylinders (here, a cylinder is a set where you fix some of the coordinates). The measure on these sequences is defined as follows. Say we fix the first $n$ coordinates, then the measure of this set is $2^{-n}$.

So for example the set of all outcomes with the first coordinate equal to $1$ is $1/2$ (the probability of getting a head). The set of all outcomes with first two coordinates $1,0$ is $1/4$. With the measure defined on cylinders it extends naturally to all sets.