  Ergodic homeomorphisms $T$ and $S$ of Polish probability spaces $X$ and $Y$
are evenly Kakutani equivalent if there is an orbit equivalence $\phi: X_0
\rightarrow Y_0$ between full measure subsets of $X$ and $Y$ such that, for
some $A \subset X_0$ of positive measure, $\phi$ restricts to a measurable
isomorphism of the induced systems $T_A$ and $S_{\phi(A)}$. The study of even
Kakutani equivalence dates back to the seventies, and it is well known that any
two zero-entropy loosely Bernoulli systems are evenly Kakutani equivalent. But
even Kakutani equivalence is a purely measurable relation, while systems such
as the Morse minimal system are both measurable and topological.
  Recently del Junco, Rudolph and Weiss studied a new relation called nearly
continuous Kakutani equivalence. A nearly continuous Kakutani equivalence is an
even Kakutani equivalence where also $X_0$ and $Y_0$ are invariant $G_\delta$
sets, $A$ is within measure zero of both open and closed, and $\phi$ is a
homeomorphism from $X_0$ to $Y_0$. It is known that nearly continuous Kakutani
equivalence is strictly stronger than even Kakutani equivalence, and nearly
continuous Kakutani equivalence is the natural strengthening of even Kakutani
equivalence to the nearly continuous category---the category where maps are
continuous after sets of measure zero are removed. In this paper we show that
the Morse minimal substitution system is nearly continuously Kakutani
equivalent to the binary odometer.
