  The problem of Shannon entropy estimation in countable infinite alphabets is
addressed from the study and use of convergence results of the entropy
functional, which is known to be discontinuous with respect to the total
variation distance in $\infty$-alphabets. Sufficient conditions for the
convergence of the entropy are used, including scenarios with both finitely and
infinitely supported assumptions on the distributions. From this new
perspective, four plug-in histogram-based estimators are studied showing that
convergence results are instrumental to derive new strong consistency and rate
of convergences results. Different scenarios and conditions are used on both
the estimators and the underlying distribution, considering for example finite
and unknown supported assumptions and summable tail bounded conditions.
