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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∞

Pantsulaia G.

Bulletin of the Polish Academy of Sciences. Mathematics

2009

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

Pantsulaia G.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

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

Pantsulaia G.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

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].