Abstract
Abstract
We consider the family
{\mathcal{CS}}
of symmetric Cantor subsets of
{[0,1]}
.
Each set in
{\mathcal{CS}}
is uniquely determined by a sequence
{a=(a_{n})}
belonging to the Polish space
{X\mathrel{\mathop{:}}=(0,1)^{\mathbb{N}}}
equipped with probability product measure μ.
This yields a one-to-one correspondence between sets in
{\mathcal{CS}}
and sequences in X.
If
{\mathcal{A}\subset\mathcal{CS}}
, the corresponding subset of X is denoted by
{\mathcal{A}^{\ast}}
.
We study the subfamilies
{\mathcal{H}_{0}}
,
{\mathcal{SP}}
and
{\mathcal{M}}
of
{\mathcal{CS}}
, consisting (respectively) of sets with Haudsdorff dimension 0, and of strongly porous and microscopic sets.
We have
{\mathcal{M}\subset\mathcal{H}_{0}\subset\mathcal{SP}}
, and these inclusions are proper.
We prove that the sets
{\mathcal{M}^{\ast}}
,
{\mathcal{H}_{0}^{\ast}}
,
{\mathcal{SP}^{\ast}}
are residual in X, and
{\mu(\mathcal{H}_{0}^{\ast})=0}
,
{\mu(\mathcal{SP}^{\ast})=1}
.
Reference28 articles.
1. Some remarks on small sets;Ric. Mat.,2001
2. Some remarks on small sets;Ric. Mat.,2001
3. General approach to microscopic-type sets;J. Math. Anal. Appl.,2018
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