Upper envelopes of inner premeasures

König, Heinz

doi:10.5802/aif.1759

Cited by 7 publications

(8 citation statements)

References 11 publications

(14 reference statements)

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“…The present section wants to derive from the earlier paper [5] two related theorems of transplantation type, which are abstract versions of the results [2, 416K and 416L] discussed above.…”

Section: Two Further Theoremsmentioning

confidence: 99%

“…The basic aim of [5] was to represent certain set functions as upper envelopes of inner premeasures. We recall a few concepts and results, as before on a fixed nonvoid set X.…”

Section: Two Further Theoremsmentioning

confidence: 99%

“…It will be seen in the final Section 4 that 416K and 416L are likewise consequences of transplantation type theorems, this time of results in [5].…”

mentioning

confidence: 99%

“…Let X denote a nonvoid set. Besides the consequences in [3] there are related results in [5]. Then an essential step forward was the transplantation theorem [6, 2.4].…”

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confidence: 99%

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Measure theory: transplantation theorems for inner premeasures

König

2010

Arch. Math.

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Abstract.The main result is a new transplantation theorem for the inner premeasures of the author, with a few related theorems. These results have basic implications for example for the construction of Radon measures. They received a certain inspiration from the treatment of Radon measures in the treatise of Fremlin on measure theory. Mathematics Subject Classification (1991). 28A12, 28C15.Keywords. Inner premeasures and their maximal inner extensions, Transplantation theorems, Radon premeasures and Radon measures.1. Introduction. The present article is part of the author's new systematization in measure and integration initiated in [3], of which the latest accounts are [8,10]. As before we concentrate on the inner version. We recall that its basic concepts are the inner • premeasures and their maximal inner • extensions(• = στ with = finite, σ = sequential, τ = nonsequential), and that its basic devices are the inner • envelopes. We shall often make free use of the concepts and results set up so far.One of the final chapters in [3] was devoted to transplantation theorems for inner premeasures. The central results were for premeasures and hence of a certain simple character: but these results often appear in combination with topological compactness or with set-theoretic • = στ compactness, and thus lead to consequences for Radon measures and in the new • = στ theories. We recall the former main theorem [3, 18.10] = [6, 2.3], which also explains the word transplantation. Let X denote a nonvoid set.

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“…The present section wants to derive from the earlier paper [5] two related theorems of transplantation type, which are abstract versions of the results [2, 416K and 416L] discussed above.…”

Section: Two Further Theoremsmentioning

confidence: 99%

“…The basic aim of [5] was to represent certain set functions as upper envelopes of inner premeasures. We recall a few concepts and results, as before on a fixed nonvoid set X.…”

Section: Two Further Theoremsmentioning

confidence: 99%

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Measure theory: transplantation theorems for inner premeasures

König

2010

Arch. Math.

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“…0) We recall the Hahn-Banach type result [11] 1.3: Let S be a lattice in X with ∅ ∈ S. Then each isotone and modular set function ϕ : S → [0, ∞] with ϕ(∅) = 0 can be extended to a content Φ : P(X) → [0, ∞] with Φ(X) = sup ϕ. 1) Let X = [0, 1[ and A = Bor(X), and α = Leb|A be the Borel-Lebesgue measure.…”

Section: )mentioning

confidence: 99%