Nowhere Dense Sets With Positive Measure
A nowhere dense set is not necessarily negligible in every sense. For example, if X is the unit interval, not only is it possible to have a dense set of Lebesgue measure zero (such as the set of rationals), but it is also possible to have a nowhere dense set with positive measure.
For one example (a variant of the Cantor set), remove from all dyadic fractions, i.e. fractions of the form a/2n in lowest terms for positive integers a and n, and the intervals around them: (a/2n − 1/22n+1, a/2n + 1/22n+1). Since for each n this removes intervals adding up to at most 1/2n+1, the nowhere dense set remaining after all such intervals have been removed has measure of at least 1/2 (in fact just over 0.535... because of overlaps) and so in a sense represents the majority of the ambient space . This set nowhere dense, as it is closed and has an empty interior: any interval (a, b) is not contained in the set since the dyadic fractions in (a, b) have been removed.
Generalizing this method, one can construct in the unit interval nowhere dense sets of any measure less than 1.
Read more about this topic: Nowhere Dense Set
Famous quotes containing the words dense, sets, positive and/or measure:
“No sleep. The sultriness pervades the air
And binds the brain—a dense oppression, such
As tawny tigers feel in matted shades,
Vexing their blood and making apt for ravage.”
—Herman Melville (1819–1891)
“There is the name and the thing; the name is a sound which sets a mark on and denotes the thing. The name is no part of the thing nor of the substance; it is an extraneous piece added to the thing, and outside of it.”
—Michel de Montaigne (1533–1592)
“As for the terms good and bad, they indicate no positive quality in things regarded in themselves, but are merely modes of thinking, or notions which we form from the comparison of things with one another. Thus one and the same thing can be at the same time good, bad, and indifferent. For instance music is good for him that is melancholy, bad for him who mourns; for him who is deaf, it is neither good nor bad.”
—Baruch (Benedict)
“Everything is good in due measure and strong sensations know not measure.”
—Anton Pavlovich Chekhov (1860–1904)