Displaying similar documents to “The Christensen measurable solutions of a generalization of the Gołąb-Schinzel functional equation”

Semicontinuous integrands as jointly measurable maps

Oriol Carbonell-Nicolau (2014)

Commentationes Mathematicae Universitatis Carolinae

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Suppose that ( X , 𝒜 ) is a measurable space and Y is a metrizable, Souslin space. Let 𝒜 u denote the universal completion of 𝒜 . For x X , let f ̲ ( x , · ) be the lower semicontinuous hull of f ( x , · ) . If f : X × Y ¯ is ( 𝒜 u ( Y ) , ( ¯ ) ) -measurable, then f ̲ is ( 𝒜 u ( Y ) , ( ¯ ) ) -measurable.

A problem with almost everywhere equality

Piotr Niemiec (2012)

Annales Polonici Mathematici

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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 .

Parametrization of Riemann-measurable selections for multifunctions of two variables with application to differential inclusions

Giovanni Anello, Paolo Cubiotti (2004)

Annales Polonici Mathematici

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We consider a multifunction F : T × X 2 E , where T, X and E are separable metric spaces, with E complete. Assuming that F is jointly measurable in the product and a.e. lower semicontinuous in the second variable, we establish the existence of a selection for F which is measurable with respect to the first variable and a.e. continuous with respect to the second one. Our result is in the spirit of [11], where multifunctions of only one variable are considered.

Method of averaging for the system of functional-differential inclusions

Teresa Janiak, Elżbieta Łuczak-Kumorek (1996)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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The basic idea of this paper is to give the existence theorem and the method of averaging for the system of functional-differential inclusions of the form ⎧ ( t ) F ( t , x t , y t ) (0) ⎨ ⎩ ( t ) G ( t , x t , y t ) (1)

Fourier analysis, linear programming, and densities of distance avoiding sets in n

Fernando Mário de Oliveira Filho, Frank Vallentin (2010)

Journal of the European Mathematical Society

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We derive new upper bounds for the densities of measurable sets in n which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the unit distance in dimensions 2 , , 24 . This gives new lower bounds for the measurable chromatic number in dimensions 3 , , 24 . We apply it to get a short proof of a variant of a recent result of Bukh which in turn generalizes theorems...

Measurable envelopes, Hausdorff measures and Sierpiński sets

Márton Elekes (2003)

Colloquium Mathematicae

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We show that the existence of measurable envelopes of all subsets of ℝⁿ with respect to the d-dimensional Hausdorff measure (0 < d < n) is independent of ZFC. We also investigate the consistency of the existence of d -measurable Sierpiński sets.

Algebraic genericity of strict-order integrability

Luis Bernal-González (2010)

Studia Mathematica

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We provide sharp conditions on a measure μ defined on a measurable space X guaranteeing that the family of functions in the Lebesgue space L p ( μ , X ) (p ≥ 1) which are not q-integrable for any q > p (or any q < p) contains large subspaces of L p ( μ , X ) (without zero). This improves recent results due to Aron, García, Muñoz, Palmberg, Pérez, Puglisi and Seoane. It is also shown that many non-q-integrable functions can even be obtained on any nonempty open subset of X, assuming that X is a topological...

Median for metric spaces

Nacereddine Belili, Henri Heinich (2001)

Applicationes Mathematicae

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We consider a Köthe space ( , | | · | | ) of random variables (r.v.) defined on the Lebesgue space ([0,1],B,λ). We show that for any sub-σ-algebra ℱ of B and for all r.v.’s X with values in a separable finitely compact metric space (M,d) such that d(X,x) ∈ for all x ∈ M (we then write X ∈ (M)), there exists a median of X given ℱ, i.e., an ℱ-measurable r.v. Y ∈ (M) such that | | d ( X , Y ) | | | | d ( X , Z ) | | for all ℱ-measurable Z. We develop the basic theory of these medians, we show the convergence of empirical medians and we give...

A pair of linear functional inequalities and a characterization of L p -norm

Dorota Krassowska, Janusz Matkowski (2005)

Annales Polonici Mathematici

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It is shown that, under some general algebraic conditions on fixed real numbers a,b,α,β, every solution f:ℝ → ℝ of the system of functional inequalities f(x+a) ≤ f(x)+α, f(x+b) ≤ f(x)+β that is continuous at some point must be a linear function (up to an additive constant). Analogous results for three other similar simultaneous systems are presented. An application to a characterization of L p -norm is given.

True preimages of compact or separable sets for functional analysts

Lech Drewnowski (2020)

Commentationes Mathematicae Universitatis Carolinae

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We discuss various results on the existence of ‘true’ preimages under continuous open maps between F -spaces, F -lattices and some other spaces. The aim of the paper is to provide accessible proofs of this sort of results for functional-analysts.

The law of large numbers and a functional equation

Maciej Sablik (1998)

Annales Polonici Mathematici

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We deal with the linear functional equation (E) g ( x ) = i = 1 r p i g ( c i x ) , where g:(0,∞) → (0,∞) is unknown, ( p , . . . , p r ) is a probability distribution, and c i ’s are positive numbers. The equation (or some equivalent forms) was considered earlier under different assumptions (cf. [1], [2], [4], [5] and [6]). Using Bernoulli’s Law of Large Numbers we prove that g has to be constant provided it has a limit at one end of the domain and is bounded at the other end.