Sliding-block coding for weakly continuous channels

For a weakly continuous stationary channel with operational sliding-block capacity
C_{s}
, it is shown that every stationary, ergodic, aperiodic source with entropy less than
C_{s}
is sliding-block transmissible over the channel. It had been shown earlier that a stationary and ergodic source is block transmissible over the channel if its entropy is less than the operational block capacity
C_{b}
. If, in addition, the channel is ergodic, it is shown that
C_{s}=C_{b}
. An example of a nonergodic weakly continuous channel is given for which
0<C_{s}<C_{b}
.