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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4See http://planetmath.org/encyclopedia/InfiniteProductMeasure.html and http://math.stackexchange.com/q/21945 – t.b. Apr 29 '12 at 08:09
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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. – Sangchul Lee Apr 29 '12 at 09:31