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Concept

Almost everywhere

A property holds almost everywhere when the points where it fails form a set of measure zero. It is the sense in which a function can equal another, or a sequence converge, while ignoring a set too thin to have any length, area or probability.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A staircase with no steps. The Cantor function drawn to several stages: a continuous non-decreasing curve from nought to one which is constant on every interval of the complement of the middle-thirds set, so its whole rise happens on a set of measure zero.

A staircase with no steps

A function that rises from nought to one, is continuous everywhere, and has derivative zero at almost every point. All of its climbing happens on a set of no length at all, which is possible because that set has uncountably many points.

analysis · Measure A set's share of a shrinking window, at three points. a point of the set x=0.33155689: density at r=1e-1 0.5633, 1e-3 0.9885, 1e-5 0.99695; the middle of a gap x=0.07812500: density at r=1e-1 0.6250, 1e-3 0.0000, 1e-5 0.00000; the end of a gap x=0.07031250: density at r=1e-1 0.6250, 1e-3 0.4971, 1e-5 0.49998

No set fills half of every interval

A fat Cantor set has length one half and contains no interval, so every interval holds some of it and some of its gaps. Look closely at any one point, though, and the window around it ends up almost all set or almost all gap. Lebesgue proved in 1910 that this is forced for every set with a length: the share of a shrinking window is 0 or 1 almost everywhere. So no set can fill exactly half of every interval, however it is built.

analysis · Measure

Named alongside it

The objects these essays reach for when they reach for this one.

Cantor setMeasureContinuityDensityDerivativeIntegralIntervalMartingaleMonotoneSelf-similarity

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