The Lebesgue Decomposition Theorem for Arbitrary Contents

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.

Prime Decomposition Theorem for Arbitrary Semigroups: General Holonomy Decomposition and Synthesis Theorem

Herein we generalize the holonomy theorem for finite semigroups (see [7]) to arbitrary semigroups, S, by embedding s^ into an infinite Zeiger wreath product, which is then expanded to an infinite iterative matrix semigroup. If S is not finite-J-above (where finite-J-above means every element has only a finite number of divisors), then S is replaced by g3, the triple Schtitzenberger product, whi...

The Lebesgue Monotone Convergence Theorem

For simplicity, we adopt the following rules: X is a non empty set, S is a σ-field of subsets of X, M is a σ-measure on S, E is an element of S, F , G are sequences of partial functions from X into R, I is a sequence of extended reals, f , g are partial functions from X to R, s1, s2, s3 are sequences of extended reals, p is an extended real number, n, m are natural numbers, x is an element of X...

Decomposition of Lebesgue Spaces