Baire functions in the spaces of countable composition

1968 ◽  
pp. 131-143
Author(s):  
V. E. Šneĭdek
Keyword(s):  
Baire Functions

Decomposing Baire functions

1988 ◽  
Vol 14 (1) ◽  
pp. 16
Author(s):  
Cichoń ◽  
Morayne ◽  
Pawlikowski ◽  
Solecki
Keyword(s):  
Baire Functions

scholarly journals Characterizations and properties of graphs of Baire functions

2018 ◽  
Vol 68 (4) ◽  
pp. 789-802
Author(s):  
Balázs Maga
Keyword(s):  
Banach Space ◽  
Topological Space ◽  
Convex Set ◽  
Vertical Section ◽  
Fréchet Space ◽  
Frechet Space ◽  
Similar Question ◽  
Baire Functions

Abstract Let X be a paracompact topological space and Y be a Banach space. In this paper, we will characterize the Baire-1 functions f : X → Y by their graph: namely, we will show that f is a Baire-1 function if and only if its graph gr(f) is the intersection of a sequence $\begin{array}{} \displaystyle (G_n)_{n=1}^{\infty} \end{array}$ of open sets in X × Y such that for all x ∈ X and n ∈ ℕ the vertical section of Gn is a convex set, whose diameter tends to 0 as n → ∞. Afterwards, we will discuss a similar question concerning functions of higher Baire classes and formulate some generalized results in slightly different settings: for example we require the domain to be a metrized Suslin space, while the codomain is a separable Fréchet space. Finally, we will characterize the accumulation set of graphs of Baire-2 functions between certain spaces.

scholarly journals Epicompletetion of archimedean l–groups and vector lattices with weak unit

1990 ◽  
Vol 48 (1) ◽  
pp. 25-56 ◽  
Author(s):  
Richard N. Ball ◽  
Anthony W. Hager
Keyword(s):  
Vector Lattice ◽  
Boolean Algebras ◽  
Weak Order ◽  
Minimal Extension ◽  
Order Unit ◽  
Weak Unit ◽  
Vector Lattices ◽  
Measurable Functions ◽  
Baire Functions

AbstractIn the category W of archimedean l–groups with distinguished weak order unit, with unitpreserving l–homorphism, let B be the class of W-objects of the form D(X), with X basically disconnected, or, what is the same thing (we show), the W-objects of the M/N, where M is a vector lattice of measurable functions and N is an abstract ideal of null functions. In earlier work, we have characterized the epimorphisms in W, and shown that an object G is epicomplete (that is, has no proper epic extension) if and only if G ∈ B. This describes the epicompletetions of a give G (that is, epicomplete objects epically containing G). First, we note that an epicompletion of G is just a “B-completion”, that is, a minimal extension of G by a B–object, that is, by a vector lattice of measurable functions modulo null functions. (C[0, 1] has 2c non-eqivalent such extensions.) Then (we show) the B–completions, or epicompletions, of G are exactly the quotients of the l–group B(Y(G)) of real-valued Baire functions on the Yosida space Y(G) of G, by σ-ideals I for which G embeds naturally in B(Y(G))/I. There is a smallest I, called N(G), and over the embedding G ≦ B(Y(G))/N(G) lifts any homorphism from G to a B–object. (The existence, though not the nature, of such a “reflective” epicompletion was first shown by Madden and Vermeer, using locales, then verified by us using properties of the class B.) There is a unique maximal (not maximum) such I, called M(Y(G)), and B(Y(G))/M(Y(G)) is the unique essentialBcompletion. There is an intermediate σ -ideal, called Z(Y(G)), and the embedding G ≦ B(y(G))/Z(Y(G)) is a σ-embedding, and functorial for σ -homomorphisms. The sistuation stands in strong analogy to the theory in Boolean algebras of free σ -algebras and σ -extensions, though there are crucial differences.

Baire Functions and Extreme Points

1972 ◽  
Vol 79 (9) ◽  
pp. 1016
Author(s):  
Lawrence G. Brown
Keyword(s):  
Extreme Points ◽  
Baire Functions

scholarly journals Space of Baire functions. I

1974 ◽  
Vol 24 (4) ◽  
pp. 47-76 ◽  
Author(s):  
J. E. Jayne
Keyword(s):  
Baire Functions

RINGS OF BAIRE FUNCTIONS ON REALCOMPACT SPACES

1985 ◽  
Vol 11 (2) ◽  
pp. 323
Author(s):  
Cater
Keyword(s):  
Baire Functions
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