  We introduce a notion of a point-wise entropy of measures (i.e local entropy)
called neutralized local entropy, and compare it with the Brin-Katok local
entropy. We show that the neutralized local entropy coincides with Brin-Katok
local entropy almost everywhere. Neutralized local entropy is computed by
measuring open sets with a relatively simple geometric description. Our proof
uses a measure density lemma for Bowen balls, and a version of a Besicovitch
covering lemma for Bowen balls. As an application, we prove a lower point-wise
dimension bound for invariant measures, complementing the previously
established bounds for upper point-wise dimension.
