Approximation of measurable periodic functions in measure by step functions

For spaces defined by a function ? of the type of modulus of continuity, we prove direct and inverse Jackson theorems for the approximation by step functions with uniform partition.     

Approximation of measurable mappings by sequences of continuous functions

Let be a completely regular Hausdorff space, a positive, finite Baire measure on , and a separable metrizable locally convex space. Suppose is a measurable mapping. Then there exists a sequence of functions in which converges to a.e. . If the function is assumed to be weakly continuous and the measure is assumed to be -smooth, then a separability condition is not needed.

     

Distances to spaces of measurable and integrable functions

Given a complete probability space and a Banach space X we establish formulas to compute the distance from a function to the spaces of strongly measurable functions and Bochner integrable functions. We study the relationship between these distances and use them to prove some quantitative counterparts of Pettis’ measurability theorem. We also give several examples showing that some of our estimates are sharp.     

A class of uniform functions and its relationship with the class of measurable functions

Borel, Lebesgue, and Hausdorff described all uniformly closed families of real-valued functions on a set T whose algebraic properties are just like those of the set of all continuous functions with respect to some open topology on T. These families turn out to be exactly the families of all functions measurable with respect to some σ-additive and multiplicative ensembles on T. The problem of describing all uniformly closed families of bounded functions whose algebraic properties are just like those of the set of all continuous bounded functions remained unsolved. In the paper, a solution of this problem is given with the help of a new class of functions that are uniform with respect to some multiplicative families of finite coverings on T. It is proved that the class of uniform functions differs from the class of measurable functions.     

Spaces of measurable functions

For a metrizable space X and a finite measure space (Ω, $\mathfrak{M}$ , µ), the space M _µ(X) of all equivalence classes (under the relation of equality almost everywhere mod µ) of $\mathfrak{M}$ -measurable functions from Ω to X, whose images are separable, equipped with the topology of convergence in measure, and some of its subspaces are studied. In particular, it is shown that M _µ(X) is homeomorphic to a Hilbert space provided µ is (nonzero) nonatomic and X is completely metrizable and has more than one point.     

A Striktpositivstellensatz for measurable functions

A weighted sums of squares decomposition of positive Borel measurable functions on a bounded Borel subset of the Euclidean space is obtained via duality from the spectral theorem for tuples of commuting self-adjoint operators. The analogous result for polynomials or certain rational functions was amply exploited during the last decade in a variety of applications. To cite this article: M. Putinar, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Fuzzy σ-algebras and fuzzy measurable functions

In this paper we first give an axiomatic definition of a fuzzy σ-algebra which is a generalisation of the family of fuzzy events considered by L.A. Zadeh [13]. The relationship between classical and fuzzy σ-algebras and between topologies and σ-algebras, in both cases, classical and fuzzy, is established in the notation of commutative diagrams. For fuzzy topologies we refer to the work of R. Lowen [8]. After a comparison between classical measurability of functions and fuzzy measurability defined in this paper we finally introduce the product of fuzzy σ-algebras.     

Pointwise compact sets of measurable functions

If X is a compact Radon measure space, and A is a pointwise compact set of real-valued measurable functions on X, then A is compact for the topology of convergence in measure (Corollary 2H). Consequently, if X_o,..., X_n are Radon measure spaces, then a separately continuous real-valued function on X_o×X₁×...×X_n is jointly measurable (Theorem 3E). If we seek to generalize this work, we encounter some undecidable problems (§4).     

On the approximation of measurable functions by continuous functions

We present a characterization of the completed Borel measure spaces for which every measurable function, with values in a separable Frechet space, is the almost everywhere limit of a sequence of continuous functions. From this characterization one can easily obtain results that have appeared recently in the literature, in a more general form. We also examine what happens when the range is a subset of an arbitrary Banach space, and show that this case often reduces to the separable case.     

On the space of measurable functions and its topology determined by the Choquet integral

Let (X,A,μ) be a finite nonadditive measure space and M be the set of all finite measurable functions on X. The topology on M, which is determined by the Choquet integral with respect to μ, is investigated. The relationship between this topology and the one determined by the Sugeno integral is examined. Some interesting examples are included.