Gritty sets and functions
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ORIGINAL RESEARCH PAPER
Gritty sets and functions John H. Riley Jr.1 Received: 23 June 2020 / Accepted: 8 October 2020 Forum D’Analystes, Chennai 2020
Abstract We define a subset of the unit interval to be gritty if it intersects every subinterval with positive measure but does not contain any subinterval. A countable sequence of pairwise disjoint gritty sets is constructed. These are then used to construct a function which is strictly increasing but has zero derivative on a set of positive measure and a function which is unbounded on a subset of positive measure of every subinterval yet has a finite integral. Keywords Measure theory Lebesgue measure Measurable functions
Mathematics Subject Classification 28 Measure and Integration
1 Introduction Although Lebesgue measure has been studied for over a century [2], the article by Ho and Zimmerman [1] shows there are still subtleties to be discovered. In that article, a finite number of disjoint, uncountable sets of real numbers is constructed, each of which is dense in the real numbers. One of these sets has measure one in the unit interval. We define such a set as gritty—it has positive measure everywhere, but does not contain any subintervals. A natural question is can the complement of a gritty set also be gritty? The answer is more than yes: we construct a sequence of gritty sets. These sets are pairwise disjoint and each set in the sequence has positive measure when restricted to any subinterval of (0, 1). We are able to use these sets to construct two functions with interesting properties. Throughout k denotes Lebesgue measure and measure means Lebesgue measure. Appropriate references are Royden [3] or Rudin [4]. & John H. Riley Jr. [email protected] 1
Department of Mathematical and Digital Sciences, Bloomsburg University of Pennsylvania, Bloomsburg, PA, USA
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J. H. Riley Jr.
2 Gritty sets Definition 2.1 A measurable subset S of (0, 1) is gritty if S does not contain any open subintervals and for every (a, b) , (0, 1), k(S \ (a, b)) [ 0. A simple example of a gritty set is the set of irrationals in (0, 1). Our motivation for the term gritty is that grit has substance but is in tiny pieces. While the irrationals in (0, 1) are gritty, the set of rationals in (0, 1) has measure 0 and isn’t gritty. This led us to ask if the complement of a gritty set can contain a gritty set. If this happens then (0, 1) = G1 [ G2 with G1 and G2 being disjoint gritty sets. This decomposition of (0, 1) would be subtle. Both G1 and G2 would be ‘‘everywhere’’ in (0, 1) with nonzero measure. To our surprise, the answer is more than yes: there is a countable collection of pairwise disjoint gritty sets. In other words, the complement of a gritty set can contain infinitely many (disjoint) gritty sets. We now construct such a sequence of gritty sets ‹W i›. Let hr n i1 n¼1 ¼ h1=2; 1=4; 3=4; 1=8; 3=8; . . .i be the sequence of dyadic numbers in (0, 1), ordered by increasing denominators (and without repetition). Explicitly, rn = (2j ?
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