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3 Bulletin of the Polish Academy of Sciences. Mathematics

3 Pantsulaia G.

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On ordinary and standard Lebesgue measures on R∞

100%

Pantsulaia G.

2009

tom Vol. 57, no 3-4

209-222

New concepts of Lebesgue measure on R∞ are proposed and some of their realizations in the ZFC theory are given. Also, it is shown that Baker's both measures [1], [2], Mankiewicz and Preiss-Tiser generators [6] and the measure of [4] are not = α-standard Lebesgue measures on R∞ for α = (1,1,...).

Relations between shy sets and sets of v[p]-measure zero in Solovay's model

100%

Pantsulaia G.

tom Vol. 52, nr 1

63-69

An example of a non-zero non-atomic translation-invariant Borel measure V[p] on the Banach space l[p] (1 is less than or equal to p is less than or equal to infinity) is constructed in Solovay's model. It is established that, for 1 is less than or equal to p is less than or equal to infinity, the condition "v[p]-almost every element of l[p] has a property P" implies that "almost every" element of l[p] (in the sense of [4]) has the property P. It is also shown that the converse is not valid.

On an invariant Borel measure in Hilbert space

100%

Pantsulaia G.

tom Vol. 52, nr 1

47-51

An example of a nonzero [sigma]-finite Borel measure [my] with everywhere dense linear manifold I[my] of admissible (in the sense of invariance) translation vectors is constructed in the Hilbert space l[2] such that [my] and any shift [my]^[alpha] of [my] by a vector [alpha] is an element of l[2] \ I[my] are neither equivalent nor orthogonal. This extends a result established in [7].