On sets with Baire property in topological spaces. (English) Zbl 1040.28001
Summary: H. Steinhaus [Fundam. Math. 1, 93-104 (1920; JFM 47.0179.02)] proved that if a set \(A\) in the line has a positive Lebesgue measure then its distance set contains an interval. He obtained even stronger forms of this result, which are concerned with mutual distances between points in an infinite sequence of sets. Similar theorems in the case we replace distance by mutual ratio were established by Bose-Majumdar. In the present paper, we endeavour to obtain some results related to sets with Baire property in locally compact topological spaces, particular cases of which yield the Baire category analogues of the above results of Steinhaus and their corresponding form for ratios by Bose-Majumdar.
MSC:
| 28A05 | Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets |
Citations:
JFM 47.0179.02References:
| [1] | N.C. Bose-Majumdar: On some properties of sets with positive measures. Annali dell Unversita di Ferrara X(1) (1962), 1-11. |
| [2] | K.P.S. Bhaskara Rao and M. Bhaskara Rao: On the difference of two second category Baire sets in a topological group. Proc. Amer. Math. Soc. 47(1) (1975), 257-258. · Zbl 0295.22003 · doi:10.2307/2040243 |
| [3] | D.K. Ganguly and S. Basu: On ratio sets of real numbers. Indian. Jour. Pure. Appl. Math. 23(1) (1993), 15-20. · Zbl 0781.28001 |
| [4] | Z. Kominek: On the sum and difference of two sets in topological vector spaces. Fund. Math. 71(2) (1971), 163-169. · Zbl 0214.37403 |
| [5] | H.I. Miller, Xenikakis and J. Polychronis: Some properties of Baire sets and sets of positive measure. Rend. Circ. Mat. Palermo (2)31 (1982), 404-414. · Zbl 0507.28001 · doi:10.1007/BF02851150 |
| [6] | J.C. Oxtoby: Measure and Category. Springer-Verlag, 1980. · Zbl 0435.28011 |
| [7] | S. Piccard: Sur les ensembles de distance. Memoires Neuchatel Universite, 1938-39. |
| [8] | W. Sander: A generalization of a theorem of S. Piccard. Proc. Amer. Math. Soc. 74(2) (1979), 281-282. · Zbl 0414.54010 · doi:10.2307/2042307 |
| [9] | H. Steinhaus: Sur les distances des points des ensembles de measure positive. Fund. math. 1 (1920), 93-104. · JFM 47.0179.02 |
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