Measurability of unions of certain dense sets

Authors:
Alberto Alonso and Javier F. Rosenblueth

Journal:
Proc. Amer. Math. Soc. **123** (1995), 2667-2675

MSC:
Primary 28A05

DOI:
https://doi.org/10.1090/S0002-9939-1995-1242071-7

MathSciNet review:
1242071

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study measurability properties of sets of the form

*C*and

*T*seem apparent for which , where

*m*denotes the Lebesgue measure in

**R**and . In fact, we prove that if

*T*is any Lebesgue measurable subset of

**R**, then no such sets can exist: no matter which

*C*we choose, if

*A*is measurable then

*mA*equals 0 or

*mT*. Moreover, if

*A*is a nonmeasurable set, the same applies to its Lebesgue outer measure. However, if we remove the condition on

*T*of being measurable, we provide an example of (nonmeasurable) sets

*C*and

*T*for which the outer measure of

*A*lies in between 0 and the outer measure of

*T*.

**[1]**Donald L. Cohn,*Measure theory*, Birkhäuser, Boston, Mass., 1980. MR**578344****[2]**Paul R. Halmos,*Measure Theory*, D. Van Nostrand Company, Inc., New York, N. Y., 1950. MR**0033869**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1995-1242071-7

Keywords:
Lebesgue measure,
Lebesgue outer measure,
unitary operators in Hilbert spaces

Article copyright:
© Copyright 1995
American Mathematical Society