Measure-theoretical isomorphism between interval and square 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.
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 
$(.b_1 b_3 b_5 \ldots,\ .b_2 b_4 b_6 \ldots) \in [0,1]^2$
See the unit square as made up by hyperbolas $yx=k$ with $k$ between 0 and 1. Consider the rectangle spanned by $(a,0)$ and $(0,b)$. Its NE vertex belongs to the hyperbola $xy=k_0$ and it can be located by its distance from the starting point $(k_0,1)$ obtained by descending along that hyperbola. Normalize such that the distance between $(k_0,1)$ and $(1,k_0)$ is $1-ab$ and call $d/(1-ab)$ the distance from $(k_0,1)$ to the NE vertex.
Send this rectangle to the interval $[d/(1-ab), d/(1-ab)+ab]$.
Does this determine a bijection from the unit square to the unit interval?