First Baire class functions in the pluri-fine topology

Fundamenta Mathematicae, 2003

Let K be a compact metric space. A real-valued function on K is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. In this paper, we study two well known ordinal indices of Baire-1 functions, the oscillation index β and the convergence index γ. It is shown that these two indices are fully compatible in the following sense : a Baire-1 function f satisfies β(f ) ≤ ω ξ 1 · ω ξ 2 for some countable ordinals ξ 1 and ξ 2 if and only if there exists a sequence of Baire-1 functions (fn) converging to f pointwise such that sup n β(fn) ≤ ω ξ 1 and γ((fn)) ≤ ω ξ 2 . We also obtain an extension result for Baire-1 functions analogous to the Tietze Extension Theorem. Finally, it is shown that if β (f ) ≤ ω ξ 1 and β (g) ≤ ω ξ 2 , then β (f g) ≤ ω ξ , where ξ = max {ξ 1 + ξ 2 , ξ 2 + ξ 1 } . These results do not assume the boundedness of the functions involved.

Topology and its Applications, 2016

We show that every Baire class one function on a countable product of metric spaces is the pointwise limit of a sequence of continuous functions, each depending only on finitely many coordinates of the argument. It is proved also that this result does not extend to higher Baire classes (with continuous functions replaced by arbitrary ones).

2017

The present thesis consists of six research papers. The first four articles deal with topics related to potential theory, Baire–one functions and its important subclasses, in particular differences of semicontinuous functions. The first paper is devoted to the stability of the Dirichlet problem for which a new criterion in terms of Poisson equation is provided. The second paper improves the recent result obtained by Lukeš et al. It shows that the classical Dirichlet solution belongs to the B1/2 subclass of Baire–one functions. A generalization of this result to the abstract context of the Choquet theory on functions spaces is provided. Finally, an abstract Dirichlet problem for the boundary condition belonging to the class of differences of semincontinuous functions is discussed. The third paper concentrates on the Lusin–Menshov property and the approximation of Baire– one and finely continuous functions by differences of semicontinuous and finely continuous functions. It provides a...

Topology and its Applications, 2013

We prove that for a topological space X, an equiconnected space Z and a Baire-one mapping g : X → Z there exists a separately continuous mapping f :

Israel Journal of Mathematics, 2018

Let U be a bounded open subset of R d , d ≥ 2 and f ∈ C(∂U). The Dirichlet solution f CU of the Dirichlet problem associated with the Laplace equation with a boundary condition f is not continuous on the closure U of U in general if U is not regular but it is always Baire-one. Let H(U) be the space of all functions continuous on the closure U and harmonic on U and F(H(U)) be the space of uniformly bounded absolutely convergent series of functions in H(U). We prove that f CU can be obtained as a uniform limit of a sequence of functions in F(H(U)). Thus f CU belongs to the subclass B 1/2 of Baire-one functions studied for example in [8]. This is not only an improvement of the result obtained in [10] but it also shows that the Dirichlet solution on the closure U can share better properties than to be only a Baire-one function. Moreover, our proof is more elementary than that in [10]. A generalization to the abstract context of simplicial function space on a metrizable compact space is provided. We conclude the paper with a brief discussion on the solvability of the abstract Dirichlet problem with a boundary condition belonging to the space of differences of bounded semicontinuous functions complementing the results obtained in [17].

Real Analysis Exchange

We will consider the classes of first return continuous, weakly first return continuous with respect to some trajectory and almost continuous functions in the sense of Stallings and we will show that for the Baire 1 functions the classes of functions mentioned above are equal. Moreover we will define the class B F 1 of strongly F-almost everywhere first return recoverable functions and shall present the relation between family B F 1 and Baire 1 functions.

We introduce the notion of $B_1$-retract and investigate the connection between $B_1$- and $H_1$-retracts.

Tatra Mountains Mathematical Publications

The paper deals with the strong porosity of some families of real functions continuous with respect to a given topology 𝒯 or 𝒜-continuous (i.e., continuous with respect to some special family 𝒜 of sets of the real line). Particularly, porosity of those families is investigated in space of the Baire 1 functions or in the space of the Baire 1 and Darboux functions.

Fine properties of Baire one functions by Udayan B. D a r j i (Louisville, Ken.), Michael J. E v a n s (Lexington, Va.), Chris F r e i l i n g (San Bernardino, Calif.) and Richard J. O' M a l l e y (Milwaukee, Wis.

Real Analysis Exchange, 2011

In the paper [3] the authors have examined functions of the Baire class 1, where the domain and the range were metric spaces. The ε − δ characterization of such functions has been proved. In this note we examine, if replacing of the condition from [3] by it's stronger version can lead us to the characterization of some subclass of B1 on the interval [0, 1].