Semicontinuous integrands as jointly measurable maps

Oriol Carbonell-Nicolau

Displaying similar documents to “Semicontinuous integrands as jointly measurable maps”

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 $F : T \times X \to 2^{E}$ , 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.

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 $2^{ℵ ₀}$ .

Fourier analysis, linear programming, and densities of distance avoiding sets in $ℝ^{n}$

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 $ℝ^{n}$ 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 $2, \dots, 24$ . This gives new lower bounds for the measurable chromatic number in dimensions $3, \dots, 24$ . 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 $ℋ^{d}$ -measurable Sierpiński sets.

CM-Selectors for pairs of oppositely semicontinuous multivalued maps with $_{p}$ -decomposable values

Hôǹg Thái Nguyêñ, Maciej Juniewicz, Jolanta Ziemińska (2001)

Studia Mathematica

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We present a new continuous selection theorem, which unifies in some sense two well known selection theorems; namely we prove that if F is an H-upper semicontinuous multivalued map on a separable metric space X, G is a lower semicontinuous multivalued map on X, both F and G take nonconvex $L_{p} (T, E)$ -decomposable closed values, the measure space T with a σ-finite measure μ is nonatomic, 1 ≤ p < ∞, $L_{p} (T, E)$ is the Bochner-Lebesgue space of functions defined on T with values in a Banach space E, F(x)...

The Christensen measurable solutions of a generalization of the Gołąb-Schinzel functional equation

Janusz Brzdęk (1996)

Annales Polonici Mathematici

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Let K denote the set of all reals or complex numbers. Let X be a topological linear separable F-space over K. The following generalization of the result of C. G. Popa [16] is proved. Theorem. Let n be a positive integer. If a Christensen measurable function f: X → K satisfies the functional equation $f (x + f {(x)}^{n} y) = f (x) f (y)$ , then it is continuous or the set x ∈ X : f(x) ≠ 0 is a Christensen zero set.

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...

Semicontinuity and continuous selections for the multivalued superposition operator without assuming growth-type conditions

Hông Thái Nguyêñ (2004)

Studia Mathematica

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Let Ω be a measure space, and E, F be separable Banach spaces. Given a multifunction $f : Ω \times E \to 2^{F}$ , denote by $N_{f} (x)$ the set of all measurable selections of the multifunction $f (\cdot, x (\cdot)) : Ω \to 2^{F}$ , s ↦ f(s,x(s)), for a function x: Ω → E. First, we obtain new theorems on H-upper/H-lower/lower semicontinuity (without assuming any conditions on the growth of the generating multifunction f(s,u) with respect to u) for the multivalued (Nemytskiĭ) superposition operator $N_{f}$ mapping some open domain G ⊂ X into $2^{Y}$ , where X and Y are...

Categoricity of theories in $L_{κ , ω}$ , when κ is a measurable cardinal. Part 2

Saharon Shelah (2001)

Fundamenta Mathematicae

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We continue the work of [2] and prove that for λ successor, a λ-categorical theory T in $L_{κ *, ω}$ is μ-categorical for every μ ≤ λ which is above the $(2^{L S (T)}) ⁺$ -beth cardinal.

Median for metric spaces

Nacereddine Belili, Henri Heinich (2001)

Applicationes Mathematicae

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We consider a Köthe space $(, | | \cdot | |_{)}$ 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 ${| | d (X, Y) | |}_{\leq} | | d (X, Z) | |$ for all ℱ-measurable Z. We develop the basic theory of these medians, we show the convergence of empirical medians and we give...

Essential norms of the Neumann operator of the arithmetical mean

Josef Král, Dagmar Medková (2001)

Mathematica Bohemica

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Let $K \subset ℝ^{m}$ ( $m \geq 2$ ) be a compact set; assume that each ball centered on the boundary $B$ of $K$ meets $K$ in a set of positive Lebesgue measure. Let $C_{0}^{(1)}$ be the class of all continuously differentiable real-valued functions with compact support in $ℝ^{m}$ and denote by $σ_{m}$ the area of the unit sphere in $ℝ^{m}$ . With each $ϕ \in C_{0}^{(1)}$ we associate the function $W_{K} ϕ (z) = \frac{1}{σ_{m}} \underset{ℝ^{m} ∖ K}{\to} \int g r a d ϕ (x) \cdot \frac{z - x}{{| z - x |}^{m}} x$ of the variable $z \in K$ (which is continuous in $K$ and harmonic in $K ∖ B$ ). $W_{K} ϕ$ depends only on the restriction ${ϕ |}_{B}$ of $ϕ$ to the boundary $B$ of $K$ . This gives rise to a linear operator $W_{K}$ ...

Borel classes of uniformizations of sets with large sections

Petr Holický (2010)

Fundamenta Mathematicae

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We give several refinements of known theorems on Borel uniformizations of sets with “large sections”. In particular, we show that a set B ⊂ [0,1] × [0,1] which belongs to $Σ ⁰_{α}$ , α ≥ 2, and which has all “vertical” sections of positive Lebesgue measure, has a $Π ⁰_{α}$ uniformization which is the graph of a $Σ ⁰_{α}$ -measurable mapping. We get a similar result for sets with nonmeager sections. As a corollary we derive an improvement of Srivastava’s theorem on uniformizations for Borel sets with $G_{δ}$ sections. ...

The Young Measure Representation for Weak Cluster Points of Sequences in M-spaces of Measurable Functions

Hôǹg Thái Nguyêñ, Dariusz Pączka (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let ⟨X,Y⟩ be a duality pair of M-spaces X,Y of measurable functions from Ω ⊂ ℝ ⁿ into $ℝ^{d}$ . The paper deals with Y-weak cluster points ϕ̅ of the sequence $ϕ (\cdot, z_{j} (\cdot))$ in X, where $z_{j} : Ω \to ℝ^{m}$ is measurable for j ∈ ℕ and $ϕ : Ω \times ℝ^{m} \to ℝ^{d}$ is a Carathéodory function. We obtain general sufficient conditions, under which, for some negligible set $A_{ϕ}$ , the integral $I (ϕ, ν_{x}) : = \int_{ℝ^{m}} ϕ (x, λ) d ν_{x} (λ)$ exists for $x \in Ω ∖ A_{ϕ}$ and $ϕ ̅ (x) = I (ϕ, ν_{x})$ on $Ω ∖ A_{ϕ}$ , where $ν = {ν_{x}}_{x \in Ω}$ is a measurable-dependent family of Radon probability measures on $ℝ^{m}$ .

On automatic boundedness of Nemytskiĭ set-valued operators

S. Rolewicz, Wen Song (1995)

Studia Mathematica

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Let X, Y be two separable F-spaces. Let (Ω,Σ,μ) be a measure space with μ complete, non-atomic and σ-finite. Let $N_{F}$ be the Nemytskiĭ set-valued operator induced by a sup-measurable set-valued function $F : Ω \times X \to 2^{Y}$ . It is shown that if $N_{F}$ maps a modular space $(N (L (Ω, Σ, μ; X)), ϱ_{N, μ})$ into subsets of a modular space $(M (L (Ω, Σ, μ; Y)), ϱ_{M, μ})$ , then $N_{F}$ is automatically modular bounded, i.e. for each set K ⊂ N(L(Ω,Σ,μ;X)) such that $r_{K} = s u p ϱ_{N, μ} (x) : x \in K < \infty$ we have $s u p ϱ_{M, μ} (y) : y \in N_{F} (K) < \infty$ .

Periodic boundary value problem of a fourth order differential inclusion

Marko Švec (1997)

Archivum Mathematicum

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The paper deals with the periodic boundary value problem (1) $L_{4} x (t) + a (t) x (t) \in F (t, x (t))$ , $t \in J = [a, b]$ , (2) $L_{i} x (a) = L_{i} x (b)$ , $i = 0, 1, 2, 3$ , where $L_{0} x (t) = a_{0} x (t)$ , $L_{i} x (t) = a_{i} (t) L_{i - 1} x (t)$ , $i = 1, 2, 3, 4$ , $a_{0} (t) = a_{4} (t) = 1$ , $a_{i} (t)$ , $i = 1, 2, 3$ and $a (t)$ are continuous on $J$ , $a (t) \geq 0$ , $a_{i} (t) > 0$ , $i = 1, 2$ , $a_{1} (t) = a_{3} (t) \cdot F (t, x) : J \times R \to$ {nonempty convex compact subsets of $R$ }, $R = (- \infty, \infty)$ . The existence of such periodic solution is proven via Ky Fan’s fixed point theorem.

Minimax theorems without changeless proportion

Liang-Ju Chu, Chi-Nan Tsai (2003)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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The so-called minimax theorem means that if X and Y are two sets, and f and g are two real-valued functions defined on X×Y, then under some conditions the following inequality holds: $i n f_{y \in Y} s u p_{x \in X} f (x, y) \leq s u p_{x \in X} i n f_{y \in Y} g (x, y)$ . We will extend the two functions version of minimax theorems without the usual condition: f ≤ g. We replace it by a milder condition: $s u p_{x \in X} f (x, y) \leq s u p_{x \in X} g (x, y)$ , ∀y ∈ Y. However, we require some restrictions; such as, the functions f and g are jointly upward, and their upper sets are connected. On the other hand, by using some...

How many normal measures can $ℵ_{ω + 1}$ carry?

Arthur W. Apter (2006)

Fundamenta Mathematicae

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We show that assuming the consistency of a supercompact cardinal with a measurable cardinal above it, it is possible for $ℵ_{ω + 1}$ to be measurable and to carry exactly τ normal measures, where $τ \geq ℵ_{ω + 2}$ is any regular cardinal. This contrasts with the fact that assuming AD + DC, $ℵ_{ω + 1}$ is measurable and carries exactly three normal measures. Our proof uses the methods of [6], along with a folklore technique and a new method due to James Cummings.