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Hi, I'm new here. I'm trying to teach myself measure theory and probability and recently wanted to find an example of a probability space (X, A, P) where X is uncountable and the sigma-algebra A is the entire power set of X. Here was my idea: let X be the set of all strings c

I wanted to prove that (X, A, P) is a probability space. I thought I found a (rather torturous) proof of this claim, but got some initial results that led me to think my proof must have gone wrong somewhere. But before I type out my whole line of reasoning, does anyone already know for sure if the above triplet satisfies the criteria of a probability space? Thanks in advance!

_{1}c_{2}c_{3}... for [tex]c_{i} \in \{ 0, 1\}[/tex], thought of as an infinite sequence of coin flips (with 1 corresponding to heads and 0 to tails), and A = X's power set. For x in X, define the n-restriction x_{n}of x as the string of the first n digits of x. Furthermore, for S in A, define S_{n}:= {x_{n}: x is in S}. Finally, define [tex]P (S) := lim_{n\rightarrow\infty} card(S_{n})/2^{n}[/tex], i.e., the probability of S is the long-term proportion of the initial segments of the members of S to all possible initial segments.I wanted to prove that (X, A, P) is a probability space. I thought I found a (rather torturous) proof of this claim, but got some initial results that led me to think my proof must have gone wrong somewhere. But before I type out my whole line of reasoning, does anyone already know for sure if the above triplet satisfies the criteria of a probability space? Thanks in advance!

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