An Abstract Formulation of the Lebesgue Decomposition Theorem

The Lebesgue decomposition theorem for arbitrary contents

The decomposition theorem named after Lebesgue asserts that certain set functions have canonical representations as sums of particular set functions called the absolutely continuous and the singular ones with respect to some fixed set function. The traditional versions are for the bounded measures with respect to some fixed measure on a σ algebra, in final form due to Hahn 1921, and for the bou...

A Version of Lebesgue Decomposition Theorem for Non-additive Measure

In this paper, Lebesgue decomposition type theorems for non-additive measure are shown under the conditions of null-additivity, converse null-additivity, weak null-additivity and σ-null-additivity, etc.. In our discussion, the monotone continuity of set function is not required.

Abstract Decomposition Theorem and Applications

DECOMPOSITION THEOREM AND APPLICATIONS RAMI GROSSBERG AND OLIVIER LESSMANN Abstract. In this paper, we prove a decomposition theorem for abstract elementary classes K with the amalgamation property, under the assumption that certain axioms regarding independence, existence of some prime models, and regular types are satisfied. This context encompasses the following: (1) K is the class of models...

Decomposition of Lebesgue Spaces

Decomposition Theorem for Abstract Elementary Classes

elementary classes (AEC) were introduced by Shelah to generate a common framework to treat those classes of models that, in spite of being non-elementary, behave similarly to elementary classes. Part of the motivation was the study of classes of models for theories in infinitary languages. As stated in the introduction, Rami Grossberg and Olivier Lessmann presented an axiomatic framework which ...