Peter A. Loeb

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We study the Bochner and Gel fand integration of Banach space valued correspondences on a general Loeb space. Though it is well known that the Lyapunov type result on the compactness and convexity of the integral of a correspondence and the Fatou type result on the preservation of upper semicontinuity by integration are in general not valid in the setting(More)
Let (X, 3, v) be an internal measure space in a denumer-ably comprehensive enlargement. The set X is a standard measure space when equipped with the smallest standard o-algebra % containing the algebra a, where the extended real-valued measure p. on % is generated by the standard part of v. If / is fl-measurable, then its standard part / is jR-measurable on(More)
It is shown that the approximating functions used to define the Bochner integral can be formed using geometrically nice sets, such as balls, from a differentiation basis. Moreover, every appropriate sum of this form will be within a preassigned ε of the integral, with the sum for the local errors also less than ε. All of this follows from the ubiquity of(More)
A general Fatou Lemma is established for a sequence of Gelfand integrable functions from a vector Loeb space to the dual of a separable Banach space or, with a weaker assumption on the sequence, a Banach lattice. A corollary sharpens previous results in the …nite dimensional setting even for the case of scalar measures. Counterexamples are presented to show(More)
This paper shows how the Lebesgue integral can be obtained as a Riemann sum and provides an extension of the Morse Covering Theorem to open sets. Let X be a finite dimensional normed space; let µ be a Radon measure on X and let Ω ⊆ X be a µ-measurable set. For λ ≥ 1, a µ-measurable set S λ (a) ⊆ X is a λ-Morse set with tag a ∈ S λ (a) if there is r > 0 such(More)
We use the insights of Robinson's nonstandard analysis as a powerful tool to extend and simplify the construction of compactifications of regular spaces. In particular, we deal with the Stone-ˇ Cech compactification and compactifications formed from topological ends. For the nonstandard extension of a metric space, the monad of a standard point x is the set(More)
In this note we use a functional approach to the integral to obtain a special case of the Keisler-Fubini theorem; the general case can be obtained with a similar proof. An immediate appUcation is the standard Fubini theorem for products of Radon measures. Similar methods give the Weil formula for quotient groups of compact Abelian groups. 1. Introduction.(More)