Abstract concentratable and reducible linear functionals and generalized functions

In the class of linear mappings of the abstract (i.e., concretized by nothing) Poincaré invariant LIS (linear involutive space) L^a_f of test functions f:X∈R⁴ⁿ→f(X)∈C¹ into a LIS L and in the class of linear continuous mappings of the abstract Poincaré invariant locally convex topological LIS T^a_f into a locally convex topological LIS we axiomatize the notions of concentration of mappings in the closed sets δ?R⁴ⁿ and reductions of mappings to the Borel sets δ?R^4a. The axiomatization is necessary both for constructing abstract axiomatic quantum field theory and for consistent introduction of the above notions in concrete classes of generalized functions whose test function spaces contain no nontrivial finite function.