Relation To Other Measures
The Borel measure agrees with the Lebesgue measure on those sets for which it is defined; however, there are many more Lebesgue-measurable sets than there are Borel measurable sets. The Borel measure is translation-invariant, but not complete.
The Haar measure can be defined on any locally compact group and is a generalization of the Lebesgue measure (Rn with addition is a locally compact group).
The Hausdorff measure is a generalization of the Lebesgue measure that is useful for measuring the subsets of Rn of lower dimensions than n, like submanifolds, for example, surfaces or curves in R³ and fractal sets. The Hausdorff measure is not to be confused with the notion of Hausdorff dimension.
It can be shown that there is no infinite-dimensional analogue of Lebesgue measure.
Read more about this topic: Lebesgue Measure
Famous quotes containing the words relation to, relation and/or measures:
“The whole point of Camp is to dethrone the serious. Camp is playful, anti-serious. More precisely, Camp involves a new, more complex relation to “the serious.” One can be serious about the frivolous, frivolous about the serious.”
—Susan Sontag (b. 1933)
“Art should exhilarate, and throw down the walls of circumstance on every side, awakening in the beholder the same sense of universal relation and power which the work evinced in the artist, and its highest effect is to make new artists.”
—Ralph Waldo Emerson (1803–1882)
“Almost everywhere we find . . . the use of various coercive measures, to rid ourselves as quickly as possible of the child within us—i.e., the weak, helpless, dependent creature—in order to become an independent competent adult deserving of respect. When we reencounter this creature in our children, we persecute it with the same measures once used in ourselves.”
—Alice Miller (20th century)