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Zbl 0644.28002
Adamski, Wolfgang
On regular extensions of contents and measures.
(English)
[J] J. Math. Anal. Appl. 127, 211-225 (1987). ISSN 0022-247X

Let ${\cal K}$ be a lattice of subsets of X. Let N(${\cal K})$ be the family of all [0,$\infty]$-valued set functions defined on ${\cal K}$ and vanishing at $\emptyset$. If $\lambda \in N({\cal K})$ we say $\lambda$ to be modular if $$\lambda (k\sb 1)+\lambda (k\sb 2) = \lambda (k\sb 1\cap k\sb 2)+\lambda (k\sb 1\cup k\sb 2)$$ and $\lambda$ is said to be supermodular if $$\lambda (k\sb 1)+\lambda (k\sb 2)\le \lambda (k\sb 1\cap k\sb 2)+\lambda (k\sb 1\cup k\sb 2).$$ It is said to be tight if $$\lambda (k\sb 2) = \lambda (k\sb 1)+Sup\{\lambda (k)\vert \quad k\in {\cal K},\quad k\subset k\sb 2-k\sb 1\}$$ for $k\sb 1,k\sb 2\in {\cal K}$ with $k\sb 1\subset k\sb 2$. Based on the concept of tight set function and the fact that every supermodular set function defined on a lattice of sets admits a tight majorant, the author generalizes extension theorems due to others including that of his own [Trans. Am. Math. Soc. 283, 353-368 (1984; Zbl 0508.28001)].
[M.K.Nayak]
MSC 2000:
*28A12 Measures and their generalizations

Keywords: tight set function; supermodular set function; tight majorant; extension theorems

Citations: Zbl 0508.28001

Cited in: Zbl 0772.28001

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