  If ${\mathcal C}\simeq 2^{\mathbb N}$ denotes the Cantor set realized as the
infinite product of two-point groups, then a folklore result says the Cantor
map from ${\mathcal C}$ into $[0,1]$ sends Haar measure to Lebesgue measure on
the interval. In fact, ${\mathcal C}$ admits many distinct topological group
structures. In this note, we show that the Haar measures induced by these
distinct group structures are share this property. We prove this by showing
that Haar measure for any group structure is the same as Haar measure induced
by a related abelian group structure. Moreover, each abelian group structure on
${\mathcal C}$ supports a natural total order that determines a map onto the
unit interval that is monotone, and hence sends intervals in ${\mathcal C}$ to
subintervals of the unit interval. Using techniques from domain theory, we show
this implies this map sends Haar measure on ${\mathcal C}$ to Lebesgue measure
on the interval, and we then use this to contract a Borel isomorphism between
any two group structures on ${\mathcal C}$.
