Uniform Continuity of Entropy Rate With Respect to the ̅F-Pseudometric

Assume that a sequence
$x=x_{0}x_{1}\ldots $
is frequency-typical for a finite-valued stationary stochastic process
$\mathbf {X}$
. We prove that the function associating to
$x$
the entropy-rate
$\bar {H}(\mathbf {X})$
of
$\mathbf {X}$
is uniformly continuous when one endows the set of all frequency-typical sequences with the
$\bar f$
pseudometric. As a consequence, we obtain the same result for the
$\bar d$
pseudometric. We also give an alternative proof of the Abramov formula for the Kolmogorov-Sinai entropy of the induced measure-preserving transformation.