Outer Measure Research Articles

Hölder Regularity of Solutions to Second-Order Elliptic Equations in Nonsmooth Domains

We establish the global Holder estimates for solutions to second-order elliptic equations, which vanish on the boundary, while the right-hand side is allowed to be unbounded. For nondivergence elliptic equations in domains satisfying an cone condition, similar results were obtained by J. H. Michael, who in turn relied on the barrier techniques due to K. Miller. Our approach is based on special growth lemmas, and it works for both divergence and nondivergence, elliptic and parabolic equations, in domains satisfying a general exterior measure condition.

Skorohod representation on a given probability space

Let \((\Omega,\mathcal{A},P)\) be a probability space, S a metric space, μ a probability measure on the Borel σ-field of S, and \(X_n:\Omega\rightarrow S\) an arbitrary map, n = 1,2,.... If μ is tight and X n converges in distribution to μ (in Hoffmann–Jørgensen’s sense), then X∼μ for some S-valued random variable X on \((\Omega,\mathcal{A},P)\). If, in addition, the X n are measurable and tight, there are S-valued random variables \(\overset{\sim}{X}_n\) and X, defined on \((\Omega,\mathcal{A},P)\), such that \(\overset{\sim}{X}_n\sim X_n\), X∼μ, and \(\overset{\sim}{X}_{n_k}\rightarrow X\) a.s. for some subsequence (n k ). Further, \(\overset{\sim}{X}_n\rightarrow X\) a.s. (without need of taking subsequences) if μ{x} = 0 for all x, or if P(X n = x) = 0 for some n and all x. When P is perfect, the tightness assumption can be weakened into separability up to extending P to \(\sigma(\mathcal{A}\cup\{H\})\) for some H⊂Ω with P *(H) = 1. As a consequence, in applying Skorohod representation theorem with separable probability measures, the Skorohod space can be taken \(((0,1),\sigma(\mathcal{U}\cup\{H\}),m_H)\), for some H⊂ (0,1) with outer Lebesgue measure 1, where \(\mathcal{U}\) is the Borel σ-field on (0,1) and m H the only extension of Lebesgue measure such that m H (H) = 1. In order to prove the previous results, it is also shown that, if X n converges in distribution to a separable limit, then X n k converges stably for some subsequence (n k ).

Mapping properties that preserve convergence in measure on finite measure spaces

Given a finite measure space ( X , M , μ ) and given metric spaces Y and Z, we prove that if { f n : X → Y | n ∈ N } is a sequence of arbitrary mappings that converges in outer measure to an M -measurable mapping f : X → Y and if g : Y → Z is a mapping that is continuous at each point of the image of f, then the sequence g ○ f n converges in outer measure to g ○ f . We must use convergence in outer measure, as opposed to (pure) convergence in measure, because of certain set-theoretic difficulties that arise when one deals with nonseparably valued measurable mappings. We review the nature of these difficulties in order to give appropriate motivation for the stated result.

EXTENSIONS OF PARTIAL LATTICE-VALUED POSSIBILISTIC MEASURES FROM NESTED DOMAINS

We investigate a partial non-numerical possibilistic measure taking its values in a complete lattice and defined on a nested system of subsets of the universe under consideration. Our aim is to extend this measure conservatively to the power-set of all systems of this universe using the same idea as that when introducing outer measures. Hence, we ascribe to each subset of the universe its minimal (in the sense of set inclusion) covering by a set from the nested domain and define its possibility degree as identical with the value ascribed to this covering. Also analyzed is the case when they are two lattice-valued possibilistic measures on the same universe, each with its own nested domain and our aim is to define one possibilistic measure on the power-set in question in a way sophistically taking profit of the information offered by both the particular partial lattice-valued measures.

Bessel capacities and rectangular differentiation in Besov spaces

We consider the differentiation of integrals of functions in Besov spaces with respect to the basis of arbitrarily oriented rectangular parallelepipeds in R n . We study almost everywhere convergence with respect to Bessel capacities. These outer measures are more sensitive than n-dimensional Lebesgue measure, and therefore we improve the positive results in [H. Aimar, L. Forzani, V. Naibo, Rectangular differentiation of integrals of Besov functions, Math. Res. Lett. 9 (2–3) (2002) 173–189].

Outer measures and weak type (1,1) estimates of Hardy-Littlewood maximal operators

We will introduce the times modified centered and uncentered Hardy-Littlewood maximal operators on nonhomogeneous spaces for. We will prove that the times modified centered Hardy-Littlewood maximal operator is weak type bounded with constant when if the Radon measure of the space has "continuity" in some sense. In the proof, we will use the outer measure associated with the Radon measure. We will also prove other results of Hardy-Littlewood maximal operators on homogeneous spaces and on the real line by using outer measures.

Body development of Simmental bull dams

Body development and production capacity in dairy production are in relation. However, body development, i.e. exterior of cows doesn?t define its functionality which is manifested in production of milk. Therefore, the capacity of cow to produce milk can only be determined more precisely by direct measuring of the production. Visual evaluation of body development and way to recognize the characteristics of dairy cows are initial parameters of milk traits, partially also longevity, as well as reproductive traits of the head of cattle, which is important from many aspects of the economical efficiency in dairy production. Exterior measures taken when cows were selected to be included in herd of bull dams, and which are also subject of this research, are following: height of withers, carcass length, breast depth, breast girth, body weight. For all investigated traits basic variation-statistical parameters were calculated: arithmetic mean value, standard deviation, coefficients of variation, standard error, variation interval. Average values of exterior measures obtained in this research were following: height to withers 136.29 cm, carcass length 165.50 cm, breast depth 72.97 cm, breast girth 200.07 cm, body weight 686.27 kg. Based on presented data it can be concluded that traits of body development of Simmental bull dams are above the average values determined for controlled population, which further justifies our conclusion that bull dams are "top" heads of cattle of the main herd.

On Measurable Sierpiński-Zygmund Functions

It is proved that there exists a Sierpinski-Zygmund function, which is measurable with respect to a certain invariant extension of the Lebesgue measure on the real line R. Let E be a nonempty set and let f : E → R be a function. We say that f is absolutely nonmeasurable if f is nonmeasurable with respect to any nonzero σ-finite diffused (i.e., continuous) measure μ defined on a σ-algebra of subsets of E. Recall that a set X ⊂ R is universal measure zero if, for every σ-finite diffused Borel measure μ on R, the equality μ∗(X) = 0 is satisfied, where μ∗ denotes, as usual, the outer measure associated with μ. It is well known that there exist uncountable universal measure zero subsets of R (see, e.g., [8]). We have a characterization of absolutely nonmeasurable functions in terms of universal measure zero sets and preimages of singletons. Theorem 1. Let f : E → R be a function. The following two assertions are equivalent: 2000 Mathematics Subject Classification. Primary: 28A05, 28D05.

Zero–infinity laws in Diophantine approximation

It is shown that for any translation invariant outer measure M, the M-measure of the intersection of any subset of Rn that is invariant under rational translations and which does not have full Lebesgue measure with the closure of an open set of positive measure cannot be positive and finite. Analogues for p-adic fields and fields of formal power series over a finite field are established. The results are applied to some problems in metric Diophantine approximation.

Almost-measurability relation induced by lattice-valued partial possibilistic measures

Possibilitic measures are set functions which can be taken into consideration as a mathematical tool for uncertainty quantification and processing, alternative to standard probability measures. For a number of practical reasons and restrictions, worth a more detailed investigating are partial possibilistic measures with non-numerical values, e.g. with values in partially ordered sets (POS) in general or in complete lattices in particular. For non-measurable sets, i.e. for the sets outside the domain of the partial possibilistic measure in question, we define their inner and outer measure, applying the basic idea of classical measure theory and approximating these sets, in the best possible way, by their measurable subsets and coverings. Inspired by the notion of symmetric difference, we introduce a lattice-valued metric or distance function and define a set to be almost measurable, if the distance between the value of its inner and outer measure is below a "small" lattice-valued threshold values, so generalizing the idea of measurability in the Lebesgue sense from the standard measure theory. A number of results dealing with the notion of lattice-valued almost-measurability and with the classes of almost measurable sets are stated and proved.

The Dual Group of a Dense Subgroup

Throughout this abstract, G is a topological Abelian group and $$\hat G$$ is the space of continuous homomorphisms from G into the circle group $${\mathbb{T}}$$ in the compact-open topology. A dense subgroup D of G is said to determine G if the (necessarily continuous) surjective isomorphism $$\hat G \to \hat D$$ given by $$h \mapsto h\left| D \right.$$ is a homeomorphism, and G is determined if each dense subgroup of G determines G. The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is determined. The authors offer several related results, including these. 1. There are (many) nonmetrizable, noncompact, determined groups. 2. If the dense subgroup D i determines G i with G i compact, then $$ \oplus _i D_i $$ determines Πi G i. In particular, if each G i is compact then $$ \oplus _i G_i $$ determines Πi G i. 3. Let G be a locally bounded group and let G + denote G with its Bohr topology. Then G is determined if and only if G + is determined. 4. Let non $$\left( {\mathcal{N}} \right)$$ be the least cardinal κ such that some $$X \subseteq {\mathbb{T}}$$ of cardinality κ has positive outer measure. No compact G with $$w\left( G \right) \geqslant non\left( {\mathcal{N}} \right)$$ is determined; thus if $$\left( {\mathcal{N}} \right) = {\mathfrak{N}}_1 $$ (in particular if CH holds), an infinite compact group G is determined if and only if w(G) = ω. Question. Is there in ZFC a cardinal κ such that a compact group G is determined if and only if w(G) < κ? Is $$\kappa = non\left( {\mathcal{N}} \right)?\kappa = {\mathfrak{N}}_1 ?$$

Paradoxical decompositions of planar sets of positive outer measure

We give examples of paradoxical subsets of the plane which do not have Lebesgue measure zero.

SUPPORT OF A JOINT RESOLUTION OF IDENTITY AND THE PROJECTION SPECTRAL THEOREM

Let A = (Ax)x ∈ Xbe a family of commuting normal operators in a separable Hilbert space H0. Obtaining the spectral expansion of A involves constructing of the corresponding joint resolution of identity E. The support supp E is not, in general, a set of full measure. This causes numerous difficulties, in particular, when proving the projection spectral theorem, i.e. the main theorem about the expansion in generalized joint eigenvectors. In this work, we show that supp E has a full outer measure under the conditions of the projection spectral theorem. Using this result, we simplify the proof of the theorem and refine its assertions.

Games of incomplete information, ergodic theory, and the measurability of equilibria

We present an example of a one-stage three-player game of incomplete information played on a sequence space {0, 1} Z such that the players’ locally finite beliefs are conditional probabilities of the canonical Bernoulli distribution on {0, 1} Z , each player has only two moves, the payoff matrix is determined by the 0-coordinate and all three players know that part of the payoff matrix pertaining to their own payoffs. For this example there are many equilibria (assuming the axiom of choice) but none that involve measurable selections of behavior by the players. By measurable we mean with respect to the completion of the canonical probability measure, e.g., all subsets of outer measure zero are measurable. This example demonstrates that the existence of equilibria is also a philosophical issue.

On finitely subadditive outer measures and modularity properties

Let ν be a finite, finitely subadditive outer measure on P(X). Define ρ (E) = ν (X) − ν (E′) for E ⊂ X. The measurable sets Sν and Sρ and the set S = {E ⊂ X/ν (E) = ρ (E)} are investigated in general, and in the presence of regularity or modularity assumptions on ν. This is also done for ν0(E) = inf{ν (M)/E ⊂ M ∈ Sν }. General properties of ν are derived when ν is weakly submodular. Applications and numerous examples are given.

Outer and inner vanishing measures and division in $H^\infty + C$

Measures on the unit circle are well studied from the view of Fourier analysis. In this paper, we investigate measures from the view of Poisson integrals and of divisibility of singular inner functions in H^\infty + C . Especially, we study singular measures which have outer and inner vanishing measures. It is given two decompositions of a singular positive measure. As applications, it is studied division theorems in H^\infty + C .

A note on upper and lower Sugeno integrals

This paper introduces upper and lower Sugeno integrals and shows that they are representable as the Sugeno integrals with respect to the outer and inner fuzzy measures, respectively.

Initial Trace of Solutions of Some Quasilinear Parabolic Equations with Absorption

We study the existence of an initial trace of nonnegative solutions of the problemδtu−∇·(∣∇u∣p−2∇u)+uq=0in QT=Ω×(0,T). We prove that the initial trace is an outer regular Borel measure which may not be locally bounded for some values of the parameters p and q. We study also the corresponding Cauchy problems with a given generalized Borel measure as initial data.

A stronger Kolmogorov zero-one law for resource-bounded measure

Resource-bounded measure has been defined on the classes E , E 2, ESPACE , E 2 SPACE , REC , and the class of all languages. It is shown here that if C is any of these classes and X is a set of languages that is closed under finite variations and has outer measure <1 in C, then X has measure 0 in C. This result strengthens Lutz's resource-bounded generalization of the classical Kolmogorov zero-one law. It also gives a useful sufficient condition for proving that a set has measure 0 in a complexity class.

On nonstandard product measure spaces

The aim of this paper is to investigate systematically the relationship between the two different types of product probability spaces based on the Loeb space construction. For any two atomless Loeb spaces, it is shown that for fixed $r \lt s$ in $[0,1]$ there exists an increasing sequence $(A_t)_{r \lt t \lt s,t\in [0,1]}$ of in the new sense product measurable sets such that $A_t$ has measure $t$ and, with respect to the usual product, the inner and outer measures are equal to $r$ and $s$, respectively. By constructing a continuum of increasing Loeb product null sets with large gaps, the Loeb product is shown to be much richer than the usual product even on null sets. General results in terms of outer and inner measures with respect to the usual product are also obtained for Loeb product measurable sets that are composed of almost independent events.