Srishti D. Chatterji

Equivalence and singularity of Gaussian measures and applications

Keywords: reproducing kernel Hilbert spaces;;; equivalence and singularity;;; Gaussian measures;;; expository paper;;; Feldman-Hajek dichotomy for Gaussian measures;;; stationary Gaussian processes;;; absolute continuity and singularity of probability measures Reference GPRO-CHAPTER-1978-003 Record created on 2010-05-25, modified on 2016-08-08

Quasi-invariance of measures under translation

Let # be a probability measure on the Bore1 field Z of the real line IR and let P = # | 1 7 4 be the product measure defined on the Borel field 2 ; ~ 1 7 6 1 7 4 1 7 4 -of the topological vector space I R ~ = I R | 1 7 4 .-. . If a = ( a , ) , e I is an element of IR ~, let P~ be the measure P translated by a i.e. P,,(A)=P(A+a) for A s Z ~. In the present paper we study the set E(P)= {aslR~~ where P ~ P , signifies that P ( A ) = 0 iff P,(A)=0. Previously E(P) had been studied by, amongst others, Dudley [4], Feldman [5], Shepp [9]. Our main result (w 3) completes theirs by showing that E(P) contains a Mazur-Orlicz vector subspace 1 ~ (for definitions see w 2) whenever it contains some 1-dimensional vector subspace and that whenever E(P) is a vector subspace then E(P)= l ~ We also give an explicit expression for the function 0. A counter-example based on one due to Dudley given in the appendix shows that E(P) need not always be a vector subspace of IR ~~ However, in w 4, we give a simple sufficient condition for E(P) to be a subspace and exhibit wide classes of examples of E(P)= l ~ for various functions 0. Our results were announced previously in [2] without proofs.

Disintegration of measures and lifting

Publisher Summary A lifting in measure theory is a choice of representatives from classes of essentially bounded measurable functions, which preserves all the usual algebraic operations. The study of the existence and uses of lifting has progressed much in the past decade mainly because of the efforts of A. and C. Ionescu–Tulcea, culminating in their monograph. Originally, the problem was solved by von Neumann in 1931 in the case of the Lebesgue measure on ℝn by using a transfinite induction process and the Lebesgue density theorem. Although the method used by von Neumann remains of considerable interest, it does not directly resolve the question of the existence of liftings for a general measure space. This was first done by D. Maharam in 1958, who used her structure theory for measure spaces to deduce a density function and then proved the existence of liftings by reverting to von Neumanns method. A fresh attack and a new solution was proposed by A. and C. Ionescu–Tulcea. This chapter presents a different arrangement of the proof using an idea of Donoghue.

Hausdorff als Maßtheoretiker

Zusammenfassung. Nach einer kurzen Beschreibung des vielseitigen mathematischen Schaffens Haussdorffs beschreiben wir speziell seine maßtheoretischen Beiträge. Aus dem unveröffentlichten Nachlass von Hausdorff stellen wir einen neuen einfachen Beweis für die Unmöglichkeit eines endlich-additiven, rotationsinvarianten und für alle Teilmengen der Einheitskugeloberfläche in

Singularity and Absolute Continuity of Measures

\mathbb{R}^3

The mathematical work of Norbert Wiener (1894-1964)

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A remark on a recent paper on the convergence of "amrts"

Publisher Summary This chapter presented how certain very simple and straight-forward considerations starting from an intuitively obvious (but technically non-trivial although well-known) supermartingale theorem lead effortlessly to a generalization of an important theorem of Kakutani and to the dichotomy theorem for Gaussian processes of Feldman and Hajek. regarding the Gaussian processes, it can be used to obtain all the usual criteria for equivalence in terms of reproducing kernel Hilbert spaces, J-divergence or suitable Hilbert-Schmidt operators.

On a theorem of Banach and Saks

Describes some parts of Wiener’s significant contribution to mathematics in as simple and non‐technical a language as possible. Looks at Wiener’s early research and how he applied integration theory to potential theory, but not without first explaining the background to integration theory. Then describes one of Wiener’s most important works – that on Brownian motion, and how other theories such as harmonic analysis flowed from his study of Brownian motion. Concludes with a brief chronology of Wiener’s life.

Some remarks on the subsequence principle in probability theory

Keywords: convergence theorem for Banach space valued qamartsq ; amarts convergence ; Radon-Nikodym property Reference GPRO-ARTICLE-1984-001 Record created on 2010-05-25, modified on 2017-08-24

A decomposition of the unit interval

In 1930, Banach and Saks [2] proved that from any bounded sequence {fn}n⩾1 of LP[0,1], 1<p<∞, a subsequence {fn(k)}n⩾1 can be so selected that it be (C,1) summable in the strong (i.e. norm) topology of LP[0,1] i.e.