Note on $s_0$ nonmeasurable unions

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Publication date: 28 May 2013

Abstract: In this note we consider an arbitrary families of sets of

s_{0}

ideal introduced by Marczewski-Szpilrajn. We show that in any uncountable Polish space

X

and under some combinatorial and set theoretical assumptions (

c o v (s_{0}) = c

for example), that for any family

c a s u b s e t e q s_{0}

with

, we can find a some subfamily

c a^{'} s u b s e t e q c a

such that the union

is not

s

-measurable. We have shown a consistency of the

c o v (s_{0}) = o m e g a_{1} < c

and existence a partition of the size

o m e g a_{1}

c a i n [s_{0}]^{o m e g a}

of the real line

, such that there exists a subfamily

c a^{'} s u b s e t e q c a

for which

is

s

-nonmeasurable. We also showed that it is relatively consistent with ZFC theory that

o m e g a_{1} < c

and existence of m.a.d. family

c a

such that

is

s

-nonmeasurable in Cantor space

2^{o} m e g a

or Baire space

o m e g a^{o} m e g a

. The consistency of

a < c o v (s_{0})

and

c o v (s_{0}) < a

is proved also.

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