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Let $(X, 𝔛, μ)$ be a standard probability space. We say that a sub-σ-algebra $𝔅$ of $𝔛$ decomposes μ in an ergodic way if any regular conditional probability $^{𝔅} P$ with respect to $𝔅$ andμ satisfies, for μ-almost every x∈X, $\forall B \in 𝔅,^{𝔅} P (x, B) \in {0, 1}$ . In this case the equality $μ (\cdot) = \int_{X}^{𝔅} P (x, \cdot) μ (d x)$ , gives us an integral decomposition in “ $𝔅$ -ergodic” components. For any sub-σ-algebra $𝔅$ of $𝔛$ , we denote by $\bar{𝔅}$ the smallest sub-σ-algebra of $𝔛$ containing $𝔅$ and the collection of all setsAin $𝔛$ satisfyingμ(A)=0. We say that $𝔅$ isμ-complete if $𝔅 = \bar{𝔅}$ . Let ${𝔅_{i} i \in I}$ be a non-empty family...

A problem of Galambos on Engel expansions

Jun Wu (2000)

Acta Arithmetica

1. Introduction. Given x in (0,1], let x = [d₁(x),d₂(x),...] denote the Engel expansion of x, that is, (1) $x = 1 / d ₁ (x) + 1 / (d ₁ (x) d ₂ (x)) + . . . + 1 / (d ₁ (x) d ₂ (x) . . . d_{n} (x)) + . . .$ , where $d_{j} (x), j \geq 1$ is a sequence of positive integers satisfying d₁(x) ≥ 2 and $d_{j + 1} (x) \geq d_{j} (x)$ for j ≥ 1. (See [3].) In [3], János Galambos proved that for almost all x ∈ (0,1], (2) $l i m_{n \to \infty} d_{n}^{1 / n} (x) = e .$ He conjectured ([3], P132) that the Hausdorff dimension of the set where (2) fails is one. In this paper, we prove this conjecture: Theorem. $d i m_{H} x \in (0, 1] : (2) f a i l s = 1$ . We use L¹ to denote the one-dimensional Lebesgue measure on (0,1] and $d i m_{H}$ to denote the Hausdorff...

A problem of invariance for Lebesgue measure

James Fickett, Jan Mycielski (1979)

Colloquium Mathematicae

A problem of von Neumann and Maharam about algebras supporting continuous submeasures

Stevo Todorcevic (2004)

Fundamenta Mathematicae

We show that a σ-algebra 𝔹 carries a strictly positive continuous submeasure if and only if 𝔹 is weakly distributive and it satisfies the σ-finite chain condition of Horn and Tarski.

A problem with almost everywhere equality

Piotr Niemiec (2012)

Annales Polonici Mathematici

A topological space Y is said to have (AEEP) if the following condition is satisfied: Whenever (X,) is a measurable space and f,g: X → Y are two measurable functions, then the set Δ(f,g) = x ∈ X: f(x) = g(x) is a member of . It is shown that a metrizable space Y has (AEEP) iff the cardinality of Y is not greater than $2^{ℵ ₀}$ .

A proof of J. Mařík's lemma

H. W. Pu, J. P. Spencer (1971)

Colloquium Mathematicae

A property of doubly stochastic densities

Anzelm Iwanik (1989)

Acta Universitatis Carolinae. Mathematica et Physica

A PU-integral on an abstract metric space

Giuseppa Riccobono (1997)

Mathematica Bohemica

In this paper, we define a -integral, i.e. an integral defined by means of partitions of unity, on a suitable compact metric measure space, whose measure $μ$ is compatible with its topology in the sense that every open set is $μ$ -measurable. We prove that the -integral is equivalent to $μ$ -integral. Moreover, we give an example of a noneuclidean compact metric space such that the above results are true.

A $Q$ -linear automorphism of the reals with non-measurable graph

Stephen Scheinberg (2019)

Commentationes Mathematicae Universitatis Carolinae

This note contains a proof of the existence of a one-to-one function $Θ$ of $ℝ$ onto itself with the following properties: $Θ$ is a rational-linear automorphism of $ℝ$ , and the graph of $Θ$ is a non-measurable subset of the plane.

A quantitative version of the isoperimetric inequality : the anisotropic case

Luca Esposito, Nicola Fusco, Cristina Trombetti (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We state and prove a stability result for the anisotropic version of the isoperimetric inequality. Namely if $E$ is a set with small anisotropic isoperimetric deficit, then $E$ is “close” to the Wulff shape set.

A Radon-Nikodym theorem for the v-integral.

S.G. Wayment, L. Hatta (1973)

Journal für die reine und angewandte Mathematik

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