  Given a partially hyperbolic diffeomorphism $f:M \rightarrow M$ defined on a
compact Riemannian manifold $M$, in this paper we define the concept of
unstable topological entropy of $f$ on a set $Y \subset M$ not necessarily
compact and we extend a theorem of R. Bowen proving that, for an ergodic
$f$-invariant measure $\mu$, the unstable measure theoretical entropy of $f$ is
upper bounded by the unstable topological entropy of $f$ on any set of full
$\mu$-measure. We also define a notion of unstable topological entropy of $f$
using a Hausdorff dimension like characterization.
