Properties of Hausdorff Measures
Note that if d is a positive integer, the d dimensional Hausdorff measure of Rd is a rescaling of usual d-dimensional Lebesgue measure which is normalized so that the Lebesgue measure of the unit cube d is 1. In fact, for any Borel set E,
where αd is the volume of the unit d-ball; it can be expressed using Euler's gamma function
Remark. Some authors adopt a definition of Hausdorff measure slightly different from the one chosen here, the difference being that it is normalized in such a way that Hausdorff d-dimensional measure in the case of Euclidean space coincides exactly with Lebesgue measure.
Read more about this topic: Hausdorff Measure
Famous quotes containing the words properties of, properties and/or measures:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“To the eyes of a god, mankind must appear as a species of bacteria which multiply and become progressively virulent whenever they find themselves in a congenial culture, and whose activity diminishes until they disappear completely as soon as proper measures are taken to sterilise them.”
—Aleister Crowley (1875–1947)