What is an explicit isomorphism between the unit interval $I = [0,1]$ with Lebesgue measure, and its square $I \times I$ with the product measure? Here isomorphism means a measure-theoretic isomorphism, which is one-one outside some set of zero measure.
Measure-theoretical isomorphism between interval and square
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measure-theory
1 Answers
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For $ x \in [0,1]$, let $x = .b_1 b_2 b_3 \ldots$ be its base-2 expansion (the choice in the ambiguous cases doesn't matter, because that's a set of measure 0). Map this to $(.b_1 b_3 b_5 \ldots,\ .b_2 b_4 b_6 \ldots) \in [0,1]^2$
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0First you need to do some work to restrict this map to full-measure subsets $X \subseteq [0,1]$ and $Y \subseteq [0,1]^2$ so that the map is well-defined and injective (note nonuniqueness of decimal expansion). Once you've made this restriction, just check that intervals get mapped to Borel subsets of $[0,1]^2$ and rectangles get mapped to Borel subsets of $[0,1]$. I need to work out the details but I think the measure-preserving calculation is relatively straightforward. – 2017-12-12