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Relations between Shy Sets and Sets of ν p -Measure Zero in Solovay’s Model

G. Pantsulaia — 2004

Bulletin of the Polish Academy of Sciences. Mathematics

An example of a non-zero non-atomic translation-invariant Borel measure ν p on the Banach space p ( 1 p ) is constructed in Solovay’s model. It is established that, for 1 ≤ p < ∞, the condition " ν p -almost every element of p has a property P" implies that “almost every” element of 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

G. Pantsulaia — 2004

Bulletin of the Polish Academy of Sciences. Mathematics

An example of a nonzero σ-finite Borel measure μ with everywhere dense linear manifold μ of admissible (in the sense of invariance) translation vectors is constructed in the Hilbert space ℓ₂ such that μ and any shift μ ( a ) of μ by a vector a μ are neither equivalent nor orthogonal. This extends a result established in [7].

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