Affiliation:
1. Department of Mathematics , Center for Advanced Study in Mathematics , Panjab University , Chandigarh , India
Abstract
Abstract
If
V
1
,
…
,
V
n
{V_{1},\dots,V_{n}}
are translations of a Vitali set by rational numbers, then we prove that
⋃
i
=
1
n
V
i
{\bigcup_{i=1}^{n}V_{i}}
contains no measurable subset of positive measure. This provides a decomposition of
ℝ
{{\mathbb{R}}}
as a countable union of disjoint sets,
any finite union of which has Lebesgue inner measure zero. As a consequence, we present a function
δ
:
ℝ
→
(
0
,
+
∞
)
{\delta:{\mathbb{R}}\rightarrow(0,+\infty)}
for which there is no measurable function
f
:
ℝ
→
ℝ
{f:{\mathbb{R}}\rightarrow{\mathbb{R}}}
satisfying
0
<
f
≤
δ
{0<f\leq\delta}
, on any measurable set of positive Lebesgue measure.
Funder
University Grants Commission, INDIA