A locally finite measure on a Hausdorff space 
is a Borel measure 
for which every point has an open set containing it with finite measure
(Teschl 2014, p. 311). Every compact set then
has finite measure, since finitely many such open
sets cover it. If 
is also locally compact,
the converse holds as well.
Local finiteness does not mean finite total measure. Lebesgue measure on 
is locally finite with infinite total measure.
In contrast, counting measure on 
is not locally finite, although each singleton
set has measure one.
