532,061 research outputs found
Closed measure zero sets
AbstractBartoszynski, T. and S. Shelah, Closed measure zero sets, Annals of Pure and Applied Logic 58 (1992) 93–110.We study the relationship between the σ-ideal generated by closed measure zero sets and the ideals of null and meager sets. We show that the additivity of the ideal of closed measure zero sets is not bigger than covering for category. As a consequence we get that the additivity of the ideal of closed measure zero sets is equal to the additivity of the ideal of meager sets
Algorithmic Randomness and Capacity of Closed Sets
We investigate the connection between measure, capacity and algorithmic
randomness for the space of closed sets. For any computable measure m, a
computable capacity T may be defined by letting T(Q) be the measure of the
family of closed sets K which have nonempty intersection with Q. We prove an
effective version of Choquet's capacity theorem by showing that every
computable capacity may be obtained from a computable measure in this way. We
establish conditions on the measure m that characterize when the capacity of an
m-random closed set equals zero. This includes new results in classical
probability theory as well as results for algorithmic randomness. For certain
computable measures, we construct effectively closed sets with positive
capacity and with Lebesgue measure zero. We show that for computable measures,
a real q is upper semi-computable if and only if there is an effectively closed
set with capacity q
Effective Capacity and Randomness of Closed Sets
We investigate the connection between measure and capacity for the space of
nonempty closed subsets of {0,1}*. For any computable measure, a computable
capacity T may be defined by letting T(Q) be the measure of the family of
closed sets which have nonempty intersection with Q. We prove an effective
version of Choquet's capacity theorem by showing that every computable capacity
may be obtained from a computable measure in this way. We establish conditions
that characterize when the capacity of a random closed set equals zero or is
>0. We construct for certain measures an effectively closed set with positive
capacity and with Lebesgue measure zero
Finding subsets of positive measure
An important theorem of geometric measure theory (first proved by Besicovitch
and Davies for Euclidean space) says that every analytic set of non-zero
-dimensional Hausdorff measure contains a closed subset of
non-zero (and indeed finite) -measure. We investigate the
question how hard it is to find such a set, in terms of the index set
complexity, and in terms of the complexity of the parameter needed to define
such a closed set. Among other results, we show that given a (lightface)
set of reals in Cantor space, there is always a
subset on non-zero -measure definable from
Kleene's . On the other hand, there are sets of reals
where no hyperarithmetic real can define a closed subset of non-zero measure.Comment: This is an extended journal version of the conference paper "The
Strength of the Besicovitch--Davies Theorem". The final publication of that
paper is available at Springer via
http://dx.doi.org/10.1007/978-3-642-13962-8_2
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