Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Completeness of sets of complex exponentials

Non-metric statistical analyses: Some metric alternatives☆

The present state as well as the historical development of problems related to the mean value property of harmonic and caloric functions are discussed. The article is based on contributions of the authors at the meeting; it is a substantially extended update of [Net2].

Mean Value Property and Harmonic Functions

We consider several elliptic boundary value problems for which there is an overspecification of data on the boundary of the domain. After reformulating the problems in an equivalent integral form, we use the alternate integral formulation to deduce that if a solution exists, then the domain must be an N-ball. Various Green's functions and classical boundary value problems of second, fourth and higher order are included among the problems considered here.

Duality theorems in some overdetermined boundary value problems

Natural choice of L1-approximants

Etude de la propriete de la moyenne des fonctions harmoniques. Existence, unicite et caracterisation de la meilleure approximation L 1 harmonique des fonctions strictement sous harmoniques

On the mean value property of harmonic functions and best harmonic ¹-approximation

Isotonic approximation in Ls

It is shown in this paper that for K-saturated models many important external sets of nonstandard analysis-such as monadic sets or the set of ail near-standard points or all pre-near-standard points or all compact points-are universally Loeb-measurable, i.e. Loeb-measurable with respect to every internal content. We furthermore obtain universal Loeb-measurability of the standard part map for topological spaces which are not covered by previous results in this . . dlrectlon. Moreover, the standard part map can be used as a measure preserving transformation for all -smooth measures, and not only for Radon-measures as known up to now. Applications of our results lead to simple new proofs for theorems of classical measure theory. We obtain e.g. the extension of -smooth Baire-measures to T-smooth Borel-measures, the decomposition theorems for -smooth Baire-measures and -smooth Borel-measures and Kakutanis theorem for product measures.

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L. Rogge

Papers

Natural choice of L1-approximants

On the mean value property of harmonic functions and best harmonic ¹-approximation

Isotonic approximation in Ls

Universal Loeb-measurability of sets and of the standard part map with applications

A note on completeness.