Search results
Results From The WOW.Com Content Network
As a particular case, a measure defined on the Euclidean space is called singular, if it is singular with respect to the Lebesgue measure on this space. For example, the Dirac delta function is a singular measure. Example. A discrete measure.
A simple example of a doubling measure is Lebesgue measure on a Euclidean space. One can, however, have doubling measures on Euclidean space that are singular with respect to Lebesgue measure. One example on the real line is the weak limit of the following sequence of measures: [ 9 ]
If T acts on Euclidean space , there is a simple geometric interpretation for the singular values: Consider the image by of the unit sphere; this is an ellipsoid, and the lengths of its semi-axes are the singular values of (the figure provides an example in ).
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean n-spaces. For lower dimensions n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume.
Such a measure is called a probability measure or distribution. See the list of probability distributions for instances. The Dirac measure δ a (cf. Dirac delta function) is given by δ a (S) = χ S (a), where χ S is the indicator function of . The measure of a set is 1 if it contains the point and 0 otherwise.
Furthermore, δ x is the only probability measure whose support is {x}. If X is n-dimensional Euclidean space R n with its usual σ-algebra and n-dimensional Lebesgue measure λ n, then δ x is a singular measure with respect to λ n: simply decompose R n as A = R n \ {x} and B = {x} and observe that δ x (A) = λ n (B) = 0. The Dirac measure ...
The measure was roundly defeated, with 98% of investors siding with the company. "Our position on these issues is not new," Costco’s board chairman Tony James said about the company’s DEI ...
Hence (singular continuous measures aside), Lebesgue decomposition gives a very explicit description of measures. The Cantor measure (the probability measure on the real line whose cumulative distribution function is the Cantor function) is an example of a singular continuous measure.