Real Analysis Exchange

Uniform Continuity of a Product of Real Functions

Sanjib Basu

Full-text: Open access


To every Lebesgue measurable subset of \(\mathbb{R}\) is associated a certain subcollection of points where the given measurable set possesses a density. By virtue of Lebesgue’s famous theorem on metric density, this associated set is a set of full measure in \(\mathbb{R}\) and is hence measure-theoretically very large. But are these sets also topologically large? In Lebesgue’s theorem, the set is kept fixed while the point is allowed to vary. If instead, we keep the point fixed a vary the set, then we may have corresponding to each point in \(\mathbb{R}\) a certain subclass of measurable sets each member of which possesses a density at that point. How large is this subclass in the “topology of measurable subsets of \(\mathbb{R}\)”? In this paper, in an endeavour to seek out answers to the questions set above, we have arrived at certain interesting and significant conclusions. Somewhat similar conclusions have been derived over analogous questions relating to ‘set-porosity’.

Article information

Real Anal. Exchange, Volume 37, Number 2 (2011), 425-438.

First available in Project Euclid: 15 April 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Primary: 26A03% 26A04
Secondary: 28A05,28A20

metric density porosity and σ-porosity meager set co-meager set Baire-property Carathéodory function Kuratowski-Ulam's theorem Lebesgue density theorem Baire-property Hausdorff metric


Basu, Sanjib. Uniform Continuity of a Product of Real Functions. Real Anal. Exchange 37 (2011), no. 2, 425--438.

Export citation


  • C.D. Aliprantis and O. Burkinshaw,Principles of Real Analysis, Harcourt Asia Ptc Ltd., 2000.
  • Ali A. Alikhani-Koopaei, Borel measurability of extreme local derivatives, Real Anal Exchange, Vol. 17 (1991-1992), 521-534.
  • C. Goffman, On Lebesgue density theorem, Proc. Amer. Math. Soc. 1 (1950), 384-387.
  • V. Kelar, Topologies generated by porosity and strong porosity, Real Anal Exchange, Vol. 16 (1990-1991), 255-267.
  • B.K Lahiri and D. Ghosh, Some properties of $\Omega$-density of sets of real numbers, Ganita (1998) 49(2), 95-100.
  • N.F.G. Martin, Lebesgue density as a set function, Pacific. J. Math. (1961), 699-784.
  • N.F.G. Martin, A note on metric density of sets of real numbers, Proc. Amer. Math. Soc. (1960), 11, 344-347.
  • J.C. Oxtoby, ph Measure and Category, Springer-Verlag, 1980.