The group-valued Lebesgue decomposition

A Lebesgue Decomposition for Elements in a Topological Group

Our aim is to establish the Lebesgue decomposition for strongly-bounded elements in a topological group. In 1963 Richard Darst established a result giving the Lebesgue decomposition of strongly-bounded elements in a normed Abelian group with respect to an algebra of projection operators. Consequently, one can establish the decomposition of strongly-bounded additive functions defined on an algeb...

Decomposition of group-valued measures on orthoalgebras

We present a general decomposition theorem for a positive inner regular finitely additive measure on an orthoalgebra L with values in an ordered topological group G, not necessarily commutative. In the case where L is a Boolean algebra, we establish the uniqueness of such a decomposition. With mild extra hypotheses on G, we extend this Boolean decomposition, preserving the uniqueness, to the ca...

Decomposition of Lebesgue Spaces

Lebesgue Integral of Simple Valued Function

The following propositions are true: (1) Let n, m be natural numbers, a be a function from [: Segn, Segm :] into R, and p, q be finite sequences of elements of R. Suppose that (i) dom p = Seg n, (ii) for every natural number i such that i ∈ dom p there exists a finite sequence r of elements of R such that dom r = Segm and p(i) = ∑ r and for every natural number j such that j ∈ dom r holds r(j) ...

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...