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Generalized projections of Borel and analytic sets

100%

Balcerzak M.

nr 1

47-53

For a σ-ideal I of sets in a Polish space X and for A ⊆ $X^2$, we consider the generalized projection 𝛷(A) of A given by 𝛷(A) = {x ∈ X: A_x ∉ I}, where $A_x$ ={y ∈ X: 〈x,y〉∈ A}. We study the behaviour of 𝛷 with respect to Borel and analytic sets in the case when I is a $∑_{2}^{0}$-supported σ-ideal. In particular, we give an alternative proof of the recent result of Kechris showing that 𝛷 [$∑_{1}^{1}(X^2)]=∑_{1}^{1}(X)$ for a wide class of $∑_{2}^{0}$-supported σ-ideals.

On weakly measurable functions

100%

Żeberski S.

tom Vol. 53, no 4

421-428

We show that if T is an uncountable Polish space, X is a metrizable space and f : T → X is a weakly Baire measurable function, then we can find a meagre set M ⊆ T such that f [T ∖ M] is a separable space. We also give an example showing that “metrizable” cannot be replaced by “normal”.

Measures in locally compact groups are carried by meager sets

63%

Zakrzewski P. , Zindulka O.

tom Vol. 7, nr 2

225-233

We show that for a σ-finite diffused Borel measure in a nondiscrete locally bounded topological group there is a meager set whose complement is of measure zero.