About: Differentiation of integrals

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In mathematics, the problem of differentiation of integrals is that of determining under what circumstances the mean value integral of a suitable function on a small neighbourhood of a point approximates the value of the function at that point. More formally, given a space X with a measure μ and a metric d, one asks for what functions f : X → R does

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rdfs:label	Differentiation of integrals (en)
rdfs:comment	In mathematics, the problem of differentiation of integrals is that of determining under what circumstances the mean value integral of a suitable function on a small neighbourhood of a point approximates the value of the function at that point. More formally, given a space X with a measure μ and a metric d, one asks for what functions f : X → R does (en)
dcterms:subject	Theorems in analysis Measure theory Theorems in calculus Differentiation rules
Wikipage page ID	13478284 (xsd:integer)
Wikipage revision ID	1117652504 (xsd:integer)
Link from a Wikipage to another Wikipage	Besicovitch covering theorem Riemann integral Countable set Covariance Mathematics Measure (mathematics) Gaussian measure Separable space Function (mathematics) Convergence in measure Locally integrable function Orthonormal basis Lebesgue differentiation theorem Lebesgue measure Locally finite measure Almost all Euclidean space Radius Henri Lebesgue Hilbert space Theorems in analysis Measure theory Theorems in calculus Dimension Average Borel measure Integral Metric space Differentiation rules Open ball Vitali covering theorem Neighbourhood (topology)
Link from a Wikipage to an external page	https://www.ams.org/journals/tran/1988-308-02/S0002-9947-1988-0951621-8/S0002-9947-1988-0951621-8.pdf
sameAs	Differentiation of integrals Differentiation of integrals Differentiation of integrals Differentiation of integrals
dbp:wikiPageUsesTemplate	dbt:Annotated_link dbt:Cite_book dbt:Cite_journal dbt:No_footnotes dbt:Calculus_topics
has abstract	In mathematics, the problem of differentiation of integrals is that of determining under what circumstances the mean value integral of a suitable function on a small neighbourhood of a point approximates the value of the function at that point. More formally, given a space X with a measure μ and a metric d, one asks for what functions f : X → R does for all (or at least μ-almost all) x ∈ X? (Here, as in the rest of the article, Br(x) denotes the open ball in X with d-radius r and centre x.) This is a natural question to ask, especially in view of the heuristic construction of the Riemann integral, in which it is almost implicit that f(x) is a "good representative" for the values of f near x. (en)
prov:wasDerivedFrom	wikipedia-en:Differentiation_of_integrals?oldid=1117652504&ns=0
page length (characters) of wiki page	6739 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Differentiation_of_integrals
is Link from a Wikipage to another Wikipage of	Leibniz integral rule Differentiation of measures Stein–Strömberg theorem
is foaf:primaryTopic of	wikipedia-en:Differentiation_of_integrals

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