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The necessary and sufficient condition for a function $f : (0, 1) \to [0, 1]$ to be Borel measurable (given by Theorem stated below) provides a technique to prove (in Corollary 2) the existence of a Borel measurable map $H : {0, 1}^{ℕ} \to {0, 1}^{ℕ}$ such that $ℒ (H (X^{p})) = ℒ (X^{1 / 2})$ holds for each $p \in (0, 1)$ , where $X^{p} = (X_{1}^{p}, X_{2}^{p}, ...)$ denotes Bernoulli sequence of random variables with $P [X_{i}^{p} = 1] = p$ .

Bocce criterion and convergence in Pettis norme in $P_{E}^{1} (μ)$ . (Critère de Bocce et convergence en norm de Pettis dans $P_{E}^{1} (μ)$ .)

Amrani, A., Bourass, A., Elamri, K. (2001)

Portugaliae Mathematica. Nova Série

Borel-approachable functions

Steven Shreve (1981)

Fundamenta Mathematicae

Cardinal algebras of functions and integration

Rolando Chuaqui (1971)

Fundamenta Mathematicae

Champs mesurables d'espaces sousliniens

Claude Delode (1977)

Annales de l'I.H.P. Probabilités et statistiques

Champs mesurables et multisections

C. Delode, O. Arino, J.-P. Penot (1976)

Annales de l'I.H.P. Probabilités et statistiques

Characterisation of weak convergence of signed measures on ...0,1...

Göran Högnäs (1977)

Mathematica Scandinavica

Characteristic functions freely generate measurable functions

Denis Higgs (1981)

Fundamenta Mathematicae

Characterizations of sum form information measures on open domains.

B.R. Ebanks, P.K. Sahoo, P. Kannappan (1997)

Aequationes mathematicae

Christensen measurability of polynomial functions and convex functions of higher orders

Zbigniew Gajda (1986)

Annales Polonici Mathematici

Closed convex hull of set of measurable functions, Riemann-measurable functions and measurability of translations

Michel Talagrand (1982)

Annales de l'institut Fourier

Let $G$ be a locally compact group. Let $L_{t}$ be the left translation in $L^{\infty} (G)$ , given by $L_{t} f (x) = f (t x)$ . We characterize (undre a mild set-theoretical hypothesis) the functions $f \in L^{\infty} (G)$ such that the map $t \to L_{t} f$ from $G$ into $L^{\infty} (G)$ is scalarly measurable (i.e. for $φ \in L^{\infty} (G)^{*}$ , $t \to φ (L_{t} f)$ is measurable). We show that it is the case when $t \to θ (L_{f} t)$ is measurable for each character $θ$ , and if $G$ is compact, if and only if $f$ is Riemann-measurable. We show that $t \to L_{t} f$ is Borel measurable if and only if $f$ is left uniformly continuous.Some of the measure-theoretic tools used there...

Compactness and Tightness in a Space of Measures with the Topology of Weak Convergence.

Flemming Topsoe (1974)

Mathematica Scandinavica

Compactness in the sense of the convergence with respect to a small system

Jacek Hejduk, Eliza Wajch (1989)

Mathematica Slovaca

Compacts de fonctions de première classe

Michèle Capon (1975/1976)

Séminaire Choquet. Initiation à l'analyse

Completeness of the set of sub- $σ$ -algebras.

Wang, Xikui (1993)

International Journal of Mathematics and Mathematical Sciences

Continuous-, derivative-, and differentiable-restrictions of measurable functions

Jack Brown (1992)

Fundamenta Mathematicae

We review the known facts and establish some new results concerning continuous-restrictions, derivative-restrictions, and differentiable-restrictions of Lebesgue measurable, universally measurable, and Marczewski measurable functions, as well as functions which have the Baire properties in the wide and restricted senses. We also discuss some known examples and present a number of new examples to show that the theorems are sharp.

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