## Real Analysis Exchange

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

### Uniform Continuity of a Product of Real Functions

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 15 April 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.rae/1366030635

**Mathematical Reviews number (MathSciNet)**

MR3016861

**Subjects**

Primary: 26A03% 26A04

Secondary: 28A05,28A20

**Keywords**

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

#### Citation

Basu, Sanjib. Uniform Continuity of a Product of Real Functions. Real Anal. Exchange 37 (2011), no. 2, 425--438. https://projecteuclid.org/euclid.rae/1366030635