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Support of a discrete measure. I have this counterexample in my mind, is it correct? For simplicity purposes I only considered a measure on the real line. Now I added some precision about the support and atoms. Ullrich Jul 19 '15 at You're exactly right, people should not use the word "support" here. More correct is to say that a discrete measure is a measure which is concentrated on a countable set.

Then this question can be closed: Ullrich for the confirmation. Sign up or log in Sign up using Google. In good cases , functions with compact support are dense in the space of functions that vanish at infinity, but this property requires some technical work to justify in a given example.

Every continuous function on a compact topological space has compact support since every closed subset of a compact space is indeed compact. In that case, the essential support of a measurable function f: For example, if f: In analysis one nearly always wants to use the essential support of a function, rather than its closed support, when the two sets are different, so ess supp f is often written simply as supp f and referred to as the support.

If M is an arbitrary set containing zero, the concept of support is immediately generalizable to functions f: Support may also be defined for any algebraic structure with identity such as a group , monoid , or composition algebra , in which the identity element assumes the role of zero. For instance, the family Z N of functions from the natural numbers to the integers is the uncountable set of integer sequences.

Functions of finite support are used in defining algebraic structures such as group rings and free abelian groups. In probability theory , the support of a probability distribution can be loosely thought of as the closure of the set of possible values of a random variable having that distribution.

There are, however, some subtleties to consider when dealing with general distributions defined on a sigma algebra , rather than on a topological space.

More formally, if X: Note that the word support can refer to the logarithm of the likelihood of a probability density function. In that example, we can consider test functions F , which are smooth functions with support not including the point 0.

Since measures including probability measures on the real line are special cases of distributions, we can also speak of the support of a measure in the same way. Then f is said to vanish on U. Hence we can define the support of f as the complement of the largest open set on which f vanishes. In Fourier analysis in particular, it is interesting to study the singular support of a distribution. This has the intuitive interpretation as the set of points at which a distribution fails to be a smooth function.

It can be expressed as an application of a Cauchy principal value improper integral. For distributions in several variables, singular supports allow one to define wave front sets and understand Huygens' principle in terms of mathematical analysis.

Singular supports may also be used to understand phenomena special to distribution theory, such as attempts to 'multiply' distributions squaring the Dirac delta function fails — essentially because the singular supports of the distributions to be multiplied should be disjoint. An abstract notion of family of supports on a topological space X , suitable for sheaf theory , was defined by Henri Cartan.

Bredon, Sheaf Theory 2nd edition, gives these definitions. If X is a locally compact space , assumed Hausdorff the family of all compact subsets satisfies the further conditions, making it paracompactifying. From Wikipedia, the free encyclopedia. This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources.

### תמיכה דיסקרטית משתינות -

By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. For example, the function f: The most common situation occurs when X is a topological סקס פרטי סרטי אורגיות such as the real line or n -dimensional Euclidean space**תמיכה דיסקרטית משתינות**f: If X is a locally compact spaceassumed Hausdorff the family of all compact subsets satisfies the further conditions, making it paracompactifying. In good casesfunctions with compact support are dense in the space of functions that vanish at infinity, but this property requires some technical work to justify in a given example. Views Read Edit View history. For distributions in several variables, singular supports allow one to define wave front sets and understand Huygens' principle in terms of mathematical analysis. Then f is said to vanish on U.

*תמיכה דיסקרטית משתינות*compactly supported smooth functions on a Euclidean space are called bump functions. This concept is used very widely in mathematical analysis. Singular supports may also be used to understand phenomena special to distribution theory, such as attempts to 'multiply' distributions squaring the Dirac delta function fails — essentially because the singular supports of the distributions to be multiplied should be disjoint.

Take a measure which atoms are exactly the rational numbers. Then the atoms are concentrated on a countable set, but its support is not countable anymore. By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service , privacy policy and cookie policy , and that your continued use of the website is subject to these policies.

Questions Tags Users Badges Unanswered. Support of a discrete measure. I have this counterexample in my mind, is it correct? For simplicity purposes I only considered a measure on the real line. Now I added some precision about the support and atoms. Ullrich Jul 19 '15 at You're exactly right, people should not use the word "support" here. More correct is to say that a discrete measure is a measure which is concentrated on a countable set.

Then this question can be closed: The most common situation occurs when X is a topological space such as the real line or n -dimensional Euclidean space and f: In this case, the support of f is defined topologically as the closure of the subset of X where f is non-zero [1] [2] [3] i.

The notion of closed support is usually applied to continuous functions, but the definition makes sense for arbitrary real or complex-valued functions on a topological space, and some authors do not require that f: For example, the function f: The condition of compact support is stronger than the condition of vanishing at infinity.

Real-valued compactly supported smooth functions on a Euclidean space are called bump functions. Mollifiers are an important special case of bump functions as they can be used in distribution theory to create sequences of smooth functions approximating nonsmooth generalized functions, via convolution.

In good cases , functions with compact support are dense in the space of functions that vanish at infinity, but this property requires some technical work to justify in a given example. Every continuous function on a compact topological space has compact support since every closed subset of a compact space is indeed compact. In that case, the essential support of a measurable function f: For example, if f: In analysis one nearly always wants to use the essential support of a function, rather than its closed support, when the two sets are different, so ess supp f is often written simply as supp f and referred to as the support.

If M is an arbitrary set containing zero, the concept of support is immediately generalizable to functions f: Support may also be defined for any algebraic structure with identity such as a group , monoid , or composition algebra , in which the identity element assumes the role of zero.

For instance, the family Z N of functions from the natural numbers to the integers is the uncountable set of integer sequences. Functions of finite support are used in defining algebraic structures such as group rings and free abelian groups. In probability theory , the support of a probability distribution can be loosely thought of as the closure of the set of possible values of a random variable having that distribution.

There are, however, some subtleties to consider when dealing with general distributions defined on a sigma algebra , rather than on a topological space. More formally, if X: Note that the word support can refer to the logarithm of the likelihood of a probability density function. In that example, we can consider test functions F , which are smooth functions with support not including the point 0.

Since measures including probability measures on the real line are special cases of distributions, we can also speak of the support of a measure in the same way. Then f is said to vanish on U. Hence we can define the support of f as the complement of the largest open set on which f vanishes. In Fourier analysis in particular, it is interesting to study the singular support of a distribution. This has the intuitive interpretation as the set of points at which a distribution fails to be a smooth function.

It can be expressed as an application of a Cauchy principal value improper integral. For distributions in several variables, singular supports allow one to define wave front sets and understand Huygens' principle in terms of mathematical analysis.