Nowhere Dense Set - Nowhere Dense Sets With Positive Measure

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:

    and Venus among the fishes skips and is a she-dolphin
    she is the gay, delighted porpoise sporting with love and the sea
    she is the female tunny-fish, round and happy among the males
    and dense with happy blood, dark rainbow bliss in the sea.
    —D.H. (David Herbert)

    The believing mind reaches its perihelion in the so-called Liberals. They believe in each and every quack who sets up his booth in the fairgrounds, including the Communists. The Communists have some talents too, but they always fall short of believing in the Liberals.
    —H.L. (Henry Lewis)

    Regna regnis lupi, The State is a wolf unto the State. It is not a pessimistic lamentation like the old homo homini lupus [Man is a wolf to Man], but a positive creed and political ideal.
    Johan Huizinga (1872–1945)

    I thought of rhyme alone,
    For rhyme can beat a measure out of trouble
    And make the daylight sweet once more....
    William Butler Yeats (1865–1939)