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35-841: (Redirected from Contents ) For a list of Misplaced Pages contents, see Misplaced Pages:Contents ; for a listing of Misplaced Pages's directories and indexes, see Misplaced Pages:Directory ; for the top-level category in Misplaced Pages's category system, see Category:Contents [REDACTED] Look up content  or contents in Wiktionary, the free dictionary. Content or contents may refer to: Media [ edit ] Content (media) , information or experience provided to audience or end-users by publishers or media producers Content industry , an umbrella term that encompasses companies owning and providing mass media and media metadata Content provider ,

70-408: A Semi ring of sets then the following statements can be deduced: If furthermore A {\displaystyle {\mathcal {A}}} is a Ring of sets one gets additionally: In general integration of functions with respect to a content does not behave well. However, there is a well-behaved notion of integration provided that the function is bounded and the total content of the space

105-500: A measure . Therefore, every (real-valued) measure is a content, but not vice versa. Contents give a good notion of integrating bounded functions on a space but can behave badly when integrating unbounded functions, while measures give a good notion of integrating unbounded functions. A classical example is to define a content on all half open intervals [ a , b ) ⊆ R {\displaystyle [a,b)\subseteq \mathbb {R} } by setting their content to

140-407: A semiring of sets in which case some additional properties can be deduced which are described below. For this reason some authors prefer to define contents only for the case of semirings or even rings. If a content is additionally σ -additive it is called a pre-measure and if furthermore A {\displaystyle {\mathcal {A}}} is a σ -algebra , the content is called

175-541: A 2011 studio album by Gang of Four Content (Joywave album) , a 2017 studio album by Joywave "Content", a 2021 song by Bo Burnham from the special Bo Burnham: Inside Periodicals [ edit ] Brill's Content , a former media watchdog publication by Steven Brill (journalist) Television and web series [ edit ] Content (web series) ; an Australian ABC comedy web series starring Charlotte Nicdao and Gemma Bird Matheson Mathematics [ edit ] Content (measure theory) ,

210-541: A 2011 studio album by Gang of Four Content (Joywave album) , a 2017 studio album by Joywave "Content", a 2021 song by Bo Burnham from the special Bo Burnham: Inside Periodicals [ edit ] Brill's Content , a former media watchdog publication by Steven Brill (journalist) Television and web series [ edit ] Content (web series) ; an Australian ABC comedy web series starring Charlotte Nicdao and Gemma Bird Matheson Mathematics [ edit ] Content (measure theory) ,

245-501: A concept in mathematics Primitive part and content , in mathematics, content is the greatest common divisor of the coefficients of a polynomial Ships [ edit ] HMS Content , ships of the British Royal Navy USS Content (SP-538) , a United States Navy vessel Other uses [ edit ] Content (Freudian dream analysis) , a dream as it is remembered and the hidden meaning of

280-416: A concept in mathematics Primitive part and content , in mathematics, content is the greatest common divisor of the coefficients of a polynomial Ships [ edit ] HMS Content , ships of the British Royal Navy USS Content (SP-538) , a United States Navy vessel Other uses [ edit ] Content (Freudian dream analysis) , a dream as it is remembered and the hidden meaning of

315-821: A functional cannot be constructed explicitly, but exists by the Hahn–Banach theorem .) Then the content of a set of positive integers is the average value of the sequence that is 1 on this set and 0 elsewhere. Informally, one can think of the content of a subset of integers as the "chance" that a randomly chosen integer lies in this subset (though this is not compatible with the usual definitions of chance in probability theory, which assume countable additivity). Frequently contents are defined on collections of sets that satisfy further constraints. In this case additional properties can be deduced that fail to hold in general for contents defined on any collections of sets. If A {\displaystyle {\mathcal {A}}} forms

350-741: A historic home located at Centreville, Maryland Content (Upper Marlboro, Maryland) also known as the Bowling House, a historic home located in Upper Marlboro, Maryland Content, Pennsylvania , an unincorporated community People with the surname [ edit ] Charles Content (born 1987), Mauritian footballer Karina Content (born 1960), Dutch writer and politician Sylvain Content (born 1971), Mauritian footballer Arts and entertainment [ edit ] Music [ edit ] Content (Gang of Four album) ,

385-433: A measure when restricted to the measurable subsets for the outer measure, which are the subsets E {\displaystyle E} such that μ ( X ) = μ ( X ∩ E ) + μ ( X ∖ E ) {\displaystyle \mu (X)=\mu (X\cap E)+\mu (X\setminus E)} for all subsets X . {\displaystyle X.} If

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420-508: A provider of non-core services in the telecommunications industry Free content , published material that can be used, copied, and modified without significant legal restriction Open content , published material licensed to authorize copying and modification by anyone Web content , information published on the World Wide Web Content analysis , a methodology used in the social sciences and humanities for studying

455-458: Is different from Wikidata All article disambiguation pages All disambiguation pages content For a list of Misplaced Pages contents, see Misplaced Pages:Contents ; for a listing of Misplaced Pages's directories and indexes, see Misplaced Pages:Directory ; for the top-level category in Misplaced Pages's category system, see Category:Contents [REDACTED] Look up content  or contents in Wiktionary,

490-431: Is different from Wikidata All article disambiguation pages All disambiguation pages Content (measure theory) That is, a content is a generalization of a measure : while the latter must be countably additive, the former must only be finitely additive. In many important applications the A {\displaystyle {\mathcal {A}}} is chosen to be a ring of sets or to be at least

525-693: Is finite, given as follows. Suppose that the total content of a space is finite. If f {\displaystyle f} is a bounded function on the space such that the inverse image of any open subset of the reals has a content, then we can define the integral of f {\displaystyle f} with respect to the content as ∫ f d λ = lim ∑ i = 1 n f ( α i ) λ ( f − 1 ( A i ) ) {\displaystyle \int f\,d\lambda =\lim \sum _{i=1}^{n}f(\alpha _{i})\lambda (f^{-1}(A_{i}))} where

560-404: The A i {\displaystyle A_{i}} form a finite collections of disjoint half-open sets whose union covers the range of f , {\displaystyle f,} and α i {\displaystyle \alpha _{i}} is any element of A i , {\displaystyle A_{i},} and where the limit is taken as

595-412: The function μ to all subsets of the topological space by μ ( A ) = inf A ⊆ U μ ( U ) . {\displaystyle \mu (A)=\inf _{A\subseteq U}\mu (U).} This is an outer measure , in other words it has the following properties: The function μ above is an outer measure on the family of all subsets. Therefore, it becomes

630-456: The Bowling House, a historic home located in Upper Marlboro, Maryland Content, Pennsylvania , an unincorporated community People with the surname [ edit ] Charles Content (born 1987), Mauritian footballer Karina Content (born 1960), Dutch writer and politician Sylvain Content (born 1971), Mauritian footballer Arts and entertainment [ edit ] Music [ edit ] Content (Gang of Four album) ,

665-437: The World Wide Web Content analysis , a methodology used in the social sciences and humanities for studying the content of communication Content format , an encoded format for converting a specific type of data to displayable information Digital content Table of contents , a list of chapters or sections in a document Places [ edit ] Content (Centreville, Maryland) also known as C.C. Harper Farm,

700-402: The compact subsets of the group, which can then be extended to a left-invariant measure. Given λ as above, we define a function μ on all open sets by μ ( U ) = sup C ⊆ U λ ( C ) . {\displaystyle \mu (U)=\sup _{C\subseteq U}\lambda (C).} This has the following properties: Given μ as above, we extend

735-424: The content is defined on all compact subsets. In general the measure is not an extension of the content, as the content may fail to be countably additive, and the measure may even be identically zero even if the content is not. First restrict the content to compact sets. This gives a function λ {\displaystyle \lambda } of compact sets C {\displaystyle C} with

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770-426: The content of communication Content format , an encoded format for converting a specific type of data to displayable information Digital content Table of contents , a list of chapters or sections in a document Places [ edit ] Content (Centreville, Maryland) also known as C.C. Harper Farm, a historic home located at Centreville, Maryland Content (Upper Marlboro, Maryland) also known as

805-412: The diameters of the sets A i {\displaystyle A_{i}} tend to 0. Suppose that μ {\displaystyle \mu } is a measure on some space X . {\displaystyle X.} The bounded measurable functions on X {\displaystyle X} form a Banach space with respect to the supremum norm. The positive elements of

840-483: The dream in Freudian analysis Contents insurance , insurance that pays for damage to, or loss of, an individual's personal possessions whilst they are located within that individual's home See also [ edit ] Content security (disambiguation) Contentment , a state of being All pages with titles beginning with Content All pages with titles containing Content Topics referred to by

875-427: The dream in Freudian analysis Contents insurance , insurance that pays for damage to, or loss of, an individual's personal possessions whilst they are located within that individual's home See also [ edit ] Content security (disambiguation) Contentment , a state of being All pages with titles beginning with Content All pages with titles containing Content Topics referred to by

910-440: The dual of this space correspond to bounded contents λ {\displaystyle \lambda } X , {\displaystyle X,} with the value of λ {\displaystyle \lambda } on f {\displaystyle f} given by the integral ∫ f d λ . {\displaystyle \int f\,d\lambda .} Similarly one can form

945-414: The following properties: There are also examples of functions λ {\displaystyle \lambda } as above not constructed from contents. An example is given by the construction of Haer measure on a locally compact group . One method of constructing such a Hear measure is to produce a left-invariant function λ {\displaystyle \lambda } as above on

980-672: The free dictionary. Content or contents may refer to: Media [ edit ] Content (media) , information or experience provided to audience or end-users by publishers or media producers Content industry , an umbrella term that encompasses companies owning and providing mass media and media metadata Content provider , a provider of non-core services in the telecommunications industry Free content , published material that can be used, copied, and modified without significant legal restriction Open content , published material licensed to authorize copying and modification by anyone Web content , information published on

1015-403: The general construction see article on Lebesgue measure . An example of a content that is not a measure on a σ -algebra is the content on all subsets of the positive integers that has value 1 / 2 n {\displaystyle 1/2^{n}} on any integer n {\displaystyle n} and is infinite on any infinite subset. An example of a content on

1050-511: The length of the intervals, that is, μ ( [ a , b ) ) = b − a . {\displaystyle \mu ([a,b))=b-a.} One can further show that this content is actually σ -additive and thus defines a pre-measure on the semiring of all half-open intervals. This can be used to construct the Lebesgue measure for the real number line using Carathéodory's extension theorem . For further details on

1085-443: The positive integers that is always finite but is not a measure can be given as follows. Take a positive linear functional on the bounded sequences that is 0 if the sequence has only a finite number of nonzero elements and takes value 1 on the sequence 1 , 1 , 1 , … , {\displaystyle 1,1,1,\ldots ,} so the functional in some sense gives an "average value" of any bounded sequence. (Such

Content - Misplaced Pages Continue

1120-502: The same term [REDACTED] This disambiguation page lists articles associated with the title Content . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Content&oldid=1189212549 " Categories : Disambiguation pages Place name disambiguation pages Disambiguation pages with surname-holder lists Hidden categories: Short description

1155-502: The same term [REDACTED] This disambiguation page lists articles associated with the title Content . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Content&oldid=1189212549 " Categories : Disambiguation pages Place name disambiguation pages Disambiguation pages with surname-holder lists Hidden categories: Short description

1190-525: The space is locally compact then every open set is measurable for this measure. The measure μ {\displaystyle \mu } does not necessarily coincide with the content λ {\displaystyle \lambda } on compact sets, However it does if λ {\displaystyle \lambda } is regular in the sense that for any compact C , {\displaystyle C,} λ ( C ) {\displaystyle \lambda (C)}

1225-442: The space of essentially bounded functions, with the norm given by the essential supremum, and the positive elements of the dual of this space are given by bounded contents that vanish on sets of measure 0. There are several ways to construct a measure μ from a content λ {\displaystyle \lambda } on a topological space. This section gives one such method for locally compact Hausdorff spaces such that

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