×

A short zero-one law proof of a result of Abian. (English) Zbl 0594.28005

Summary: In this note a new and very short zero-one law proof of the following theorem of Abian is presented. The subset of the unit interval [0,1) consisting of those real numbers whose Hamel expansions do not use a given basis element of a prescribed Hamel basis, has outer Lebesgue measure one and inner measure zero.

MSC:

28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
28A12 Contents, measures, outer measures, capacities
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Abian, A.,The outer and inner measure of a nonmeasurable set. Boll. Un. Mat. Ital. (4)3 (1970), 555–558. · Zbl 0197.33302
[2] Munroe, M. E.,Introduction to measure and integration. Addison-Wesley, Cambridge, Mass., 1953. · Zbl 0050.05603
[3] Oxtoby, J. C.,Measure and category. A survey of the analogies between topological and measure spaces. Graduate Texts in Mathematics, Vol. 2. Springer, New York-Berlin, 1971. · Zbl 0217.09201
[4] Sierpinski, W.,Sur la question de la mesurabilité de la base de M. Hamel. Fund. Math.1 (1920), 105–111. · JFM 47.0180.03
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.