I want to formally define a uniform probability measure on $\{0,1\}^\omega$ (infinite binary sequences). Is it possible? What is the exact definition? I see it should have the property "the set of sequences with exactly $k$ fixed values has probability $\frac{1}{2^k}$", but it's not a definition...
Uniform probability measure on $\{0,1\}^\omega$
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probability
sequences-and-series
measure-theory
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1Kolmogorov's theorems provide a formal construction. But by identifying $\{0,1\}^{\omega}$ with $[0, 1]$ using binary representation as usual, we immediately obtain a uniform measure inheritted from the Lebesgue measure. – 2012-04-29