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1
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Algebras of Borel measurable functions

100%
Morayne M.
Fundamenta Mathematicae
|
1992
|
tom 141
|
nr 3
229-242
EN
We determine the size levels for any function on the hyperspace of an arc as follows. Assume Z is a continuum and consider the following three conditions: 1) Z is a planar AR; 2) cut points of Z have component number two; 3) any true cyclic element of Z contains at most two cut points of Z. Then any size level for an arc satisfies 1)-3) and conversely, if Z satisfies 1)-3), then Z is a diameter level for some arc.
2
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Sierpiński's hierarchy and locally Lipschitz functions

100%
Morayne M.
Fundamenta Mathematicae
|
1995
|
tom 147
|
nr 1
73-82
EN
Let Z be an uncountable Polish space. It is a classical result that if I ⊆ ℝ is any interval (proper or not), f: I → ℝ and $α < ω_1$ then f ○ g ∈ $B_α(Z)$ for every $g ∈ B_α(Z) ∩^ZI$ if and only if f is continuous on I, where $B_α(Z)$ stands for the αth class in Baire's classification of Borel measurable functions. We shall prove that for the classes $S_α(Z) (α > 0)$ in Sierpiński's classification of Borel measurable functions the analogous result holds where the condition that f is continuous is replaced by the condition that f is locally Lipschitz on I (thus it holds for the class of differences of semicontinuous functions, which is the class $S_1(Z)$). This theorem solves the problem raised by the work of Lindenbaum ([L] and [L, Corr.]) concerning the class of functions not leading outside $S_α(Z)$ by outer superpositions.
3
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An abstract version of Sierpiński's theorem and the algebra generated by A and CA functions

64%
Cichoń J., Morayne M.
Fundamenta Mathematicae
|
1993
|
tom 142
|
nr 3
263-268
EN
We give an abstract version of Sierpiński's theorem which says that the closure in the uniform convergence topology of the algebra spanned by the sums of lower and upper semicontinuous functions is the class of all Baire 1 functions. Later we show that a natural generalization of Sierpiński's result for the uniform closure of the space of all sums of A and CA functions is not true. Namely we show that the uniform closure of the space of all sums of A and CA functions is a proper subclass of the space of all functions measurable with respect to the least class containing intersections of analytic and coanalytic sets and which is closed under countable unions (A and CA functions are analogues of lower and upper semicontinuous functions, respectively, when measurability with respect to open sets is replaced by that with respect to analytic sets).
4
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Measure preserving analytic diffeomorphisms of countable dense sets in $C^n$ and $R^n$

50%
Morayne M.
Colloquium Mathematicum
|
1987
|
tom 52
|
nr 1
93-98
5
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On differentiability of Peano type functions

38%
Morayne M.
Colloquium Mathematicum
|
1987
|
tom 53
|
nr 1
129-132
6
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On differentiability of Peano type functions. II

38%
Morayne M.
Colloquium Mathematicum
|
1987
|
tom 53
|
nr 1
133-135
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