Lusin sets

U-Lusin Sets in Hyperfinite Time Lines

In an !1{saturated nonstandard universe a cut is an initial segment of the hyperintegers, which is closed under addition. Keisler and Leth in [KL] introduced, for each given cut U , a corresponding U{topology on the hyperintegers by letting O be U open if for any x 2 O there is a y greater than all the elements in U such that the interval [x y; x+y] O. Let U be a cut in a hypernite time line H,...

On a Theorem of Banach and Kuratowski and K-lusin Sets

In a paper of 1929, Banach and Kuratowski proved—assuming the continuum hypothesis—a combinatorial theorem which implies that there is no nonvanishing σ-additive finite measure μ on R which is defined for every set of reals. It will be shown that the combinatorial theorem is equivalent to the existence of a K-Lusin set of size 20 and that the existence of such sets is independent of ZFC + ¬CH.

Lusin type theorems for Radon measures

We add to the literature the following observation. If μ is a singular measure on R which assigns measure zero to every porous set and f : R → R is a Lipschitz function which is non-differentiable μ-a.e., then for every C function g : R → R it holds μ{x ∈ Rn : f(x) = g(x)} = 0. In other words the Lusin type approximation property of Lipschitz functions with C functions does not hold with respec...

A Quantitative Lusin Theorem for Functions in BV

We extend to the BV case a measure theoretic lemma previously proved by DiBenedetto, Gianazza and Vespri ([1]) in W 1,1 loc . It states that if the set where u is positive occupies a sizable portion of a open set E then the set where u is positive clusters about at least one point of E. In this note we follow the proof given in the Appendix of [3] so we are able to use only a 1−dimensional Poin...

A constructive version of the Lusin Separation Theorem