Abstract
This present paper is concerned with set functions related to {0, 1} two valued measures. These set functions are either outer measures or have many of the same characteristics. We investigate their properties and look at relations among them. We note in particular their association with the semi‐separation of lattices.To be more specific, we define three set functions μ″, μ′, and related to μ ϵ I(L) the {0, 1} two valued set functions defined on the algebra generated by the lattice of sets L st μ is a finitely additive monotone set function for which μ(ϕ) = 0. We note relations among them and properties they possess.ln particular necessary and sufficient conditions are given for the semi‐separation of lattices in terms of equality of set functions over a lattice of subsets.Finally the notion of I‐lattice is defined, we look at some properties of these with certain other side conditions assume, and end with an application involving semi‐separation and I‐lattices.
Highlights
The notion of I-lattice is defined, we look at some properties of these with certain other side conditions assume, and end with an application involving semi-separation and I-lattices
To be more precise let X be an abstract set and L a lattice of sets containing X and for IteI(I...), the two valued {0, finitely additive non-trivial measures defined on A(L) the algebra generated by the lattice L, we define It’, and note that it is a finitely subadditive "outer measure". (See section 2 for notations and terminology, sections 3 for definitions of It’, It".) We prove that a) If L is regular
First we look at at It" which is genuine countably subadditive outer measure and is defined for all l.tel(o*, L).We define It’ which is finitely subadditive "outer measure" defined for Itel(L).We investigate some of the properties of these set functions and relationships that hold for them.We consider conditions for one lattice to semi-separate another in terms of It’ and 12 another related set function
Summary
In this paper we consider set functions that are related to a measure it, namely, it’, it", It* and . some associated premeasures.We will investigate some of their properties and look at relations among them, and note in particular their association with semi-separation.To be more precise let X be an abstract set and L a lattice of sets containing X and for IteI(I...), the two valued {0, finitely additive non-trivial measures defined on A(L) the algebra generated by the lattice L, we define It’, and note that it is a finitely subadditive "outer measure". (See section 2 for notations and terminology, sections 3 for definitions of It’, It".) We prove that a) If L is regularS(It)=S(It’).b) SIt’={E X_E and either E_L or E’_L where It(L)=l for LeL} where SIt’ are It’-measurable sets.c) LIt={ LeL it(L)=it’(L)} is a lattice, d) If Itl, it2el(L) and it l
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