  Given a general source with countably infinite source alphabet and a general
channel with arbitrary abstract channel input/channel output alphabets, we
study the joint source-channel coding problem from the information-spectrum
point of view. First, we generalize Feinstein's lemma (direct part) and
Verdu-Han's lemma (converse part) so as to be applicable to the general joint
source-channel coding problem. Based on these lemmas, we establish a sufficient
condition as well as a necessary condition for the source to be reliably
transmissible over the channel with asymptotically vanishing probability of
error. It is shown that our sufficient condition is equivalent to the
sufficient condition derived by Vembu, Verdu and Steinberg, whereas our
necessary condition is shown to be stronger than or equivalent to the necessary
condition derived by them. It turns out, as a direct consequence, that
separation principle in a relevantly generalized sense holds for a wide class
of sources and channels, as was shown in a quite dfifferent manner by Vembu,
Verdu and Steinberg. It should also be remarked that a nice duality is found
between our necessary and sufficient conditions, whereas we cannot fully enjoy
such a duality between the necessary condition and the sufficient condition by
Vembu, Verdu and Steinberg. In addition, we demonstrate a sufficient condition
as well as a necessary condition for the epsilon-transmissibility. Finally, the
separation theorem of the traditional standard form is shown to hold for the
class of sources and channels that satisfy the semi-strong converse property.
