What are the known measures defined on some sigma algebra on infinite product space of $[0,1]$ ? Is there any measure which is compatible with usual metric space structure of infinite product of $[0,1]$ ? Are these measures are significant?
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You can construct useful measures on any product of $[0,1]$, even uncountable products. Real and Abstract Analysis by Hewitt and Stromberg has such a construction. If $X_i$ is a sequence of indpendent random variables with uniform distributuion on $[0,1]$ then their joint distribution is a product measure on $[0,1]^{\infty}$. This measure is defined on the Borel sigma algebra for the metric $d$ defined by $d((x_n),(y_n))=\sum_n 2^{-n}\frac {|x_n-y_n|} {1+|x_n-y_n|}$. Such measures are very important in Probability Theory.