  In this note besides two abstract versions of the Vitali Covering Lemma an
abstract Hardy-Littlewood Maximal Inequality, generalizing weak type (1,1)
maximal function inequality, associated to any outer measure and a family of
subsets on a set is introduced. The inequality is (effectively) satisfied if
and only if a special numerical constant called Hardy-Littelwood maximal
constant is finite. Two general sufficient conditions for the finiteness of
this constant are given and upper bounds for this constant associated to the
family of (centered) balls in homogeneous spaces, family of dyadic cubes in
Euclidean spaces, family of admissible trapezoids in homogeneous trees, and
family of Calder\'{o}n-Zygmund sets in (ax+b)-group, are considered. Also a
very simple application to find some nontrivial estimates about mass density in
Classical Mechanics is given.
