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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...

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Kolmogorov'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. –  sos440 Apr 29 '12 at 9:31

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