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| In [[mathematics]], the problem of '''differentiation of integrals''' is that of determining under what circumstances the [[average|mean value]] [[integral]] of a suitable [[function (mathematics)|function]] on a small [[neighbourhood (topology)|neighbourhood]] of a point approximates the value of the function at that point. More formally, given a space ''X'' with a [[measure (mathematics)|measure]] ''μ'' and a [[metric space|metric]] ''d'', one asks for what functions ''f'' : ''X'' → '''R''' does
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| :<math>\lim_{r \to 0} \frac1{\mu \big( B_{r} (x) \big)} \int_{B_{r} (x)} f(y) \, \mathrm{d} \mu(y) = f(x)</math>
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| for all (or at least ''μ''-[[almost all]]) ''x'' ∈ ''X''? (Here, as in the rest of the article, ''B''<sub>''r''</sub>(''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''.
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| ==Theorems on the differentiation of integrals==
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| ===Lebesgue measure===
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| One result on the differentiation of integrals is the [[Lebesgue differentiation theorem]], as proved by [[Henri Lebesgue]] in 1910. Consider ''n''-[[dimension]]al [[Lebesgue measure]] ''λ''<sup>''n''</sup> on ''n''-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup>. Then, for any [[locally integrable function]] ''f'' : '''R'''<sup>''n''</sup> → '''R''', one has
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| :<math>\lim_{r \to 0} \frac1{\lambda^{n} \big( B_{r} (x) \big)} \int_{B_{r} (x)} f(y) \, \mathrm{d} \lambda^{n} (y) = f(x)</math>
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| for ''λ''<sup>''n''</sup>-almost all points ''x'' ∈ '''R'''<sup>''n''</sup>. It is important to note, however, that the measure zero set of "bad" points depends on the function ''f''.
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| ===Borel measures on '''R'''<sup>''n''</sup>===
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| The result for Lebesgue measure turns out to be a special case of the following result, which is based on the [[Besicovitch covering theorem]]: if ''μ'' is any [[locally finite measure|locally finite]] [[Borel measure]] on '''R'''<sup>''n''</sup> and ''f'' : '''R'''<sup>''n''</sup> → '''R''' is locally integrable with respect to ''μ'', then
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| :<math>\lim_{r \to 0} \frac1{\mu \big( B_{r} (x) \big)} \int_{B_{r} (x)} f(y) \, \mathrm{d} \mu (y) = f(x)</math>
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| for ''μ''-almost all points ''x'' ∈ '''R'''<sup>''n''</sup>.
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| ===Gaussian measures===
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| The problem of the differentiation of integrals is much harder in an infinite-dimensional setting. Consider a [[separable space|separable]] [[Hilbert space]] (''H'', ⟨ , ⟩) equipped with a [[Gaussian measure]] ''γ''. As stated in the article on the [[Vitali covering theorem]], the Vitali covering theorem fails for Gaussian measures on infinite-dimensional Hilbert spaces. Two results of David Preiss (1981 and 1983) show the kind of difficulties that one can expect to encounter in this setting:
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| * There is a Gaussian measure ''γ'' on a separable Hilbert space ''H'' and a Borel set ''M'' ⊆ ''H'' so that, for ''γ''-almost all ''x'' ∈ ''H'',
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| ::<math>\lim_{r \to 0} \frac{\gamma \big( M \cap B_{r} (x) \big)}{\gamma \big( B_{r} (x) \big)} = 1.</math>
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| * There is a Gaussian measure ''γ'' on a separable Hilbert space ''H'' and a function ''f'' ∈ ''L''<sup>1</sup>(''H'', ''γ''; '''R''') such that
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| ::<math>\lim_{r \to 0} \inf \left\{ \left. \frac1{\gamma \big( B_{s} (x) \big)} \int_{B_{s} (x)} f(y) \, \mathrm{d} \gamma(y) \right| x \in H, 0 < s < r \right\} = + \infty.</math>
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| However, there is some hope if one has good control over the [[covariance]] of ''γ''. Let the covariance operator of ''γ'' be ''S'' : ''H'' → ''H'' given by
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| :<math>\langle Sx, y \rangle = \int_{H} \langle x, z \rangle \langle y, z \rangle \, \mathrm{d} \gamma(z),</math>
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| or, for some [[countable set|countable]] [[orthonormal basis]] (''e''<sub>''i''</sub>)<sub>''i''∈'''N'''</sub> of ''H'',
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| :<math>Sx = \sum_{i \in \mathbf{N}} \sigma_{i}^{2} \langle x, e_{i} \rangle e_{i}.</math> | |
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| In 1981, Preiss and Jaroslav Tišer showed that if there exists a constant 0 < ''q'' < 1 such that
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| :<math>\sigma_{i + 1}^{2} \leq q \sigma_{i}^{2},</math>
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| then, for all ''f'' ∈ ''L''<sup>1</sup>(''H'', ''γ''; '''R'''),
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| :<math>\frac1{\mu \big( B_{r} (x) \big)} \int_{B_{r} (x)} f(y) \, \mathrm{d} \mu(y) \xrightarrow[r \to 0]{\gamma} f(x),</math>
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| where the convergence is [[convergence in measure]] with respect to ''γ''. In 1988, Tišer showed that if
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| :<math>\sigma_{i + 1}^{2} \leq \frac{\sigma_{i}^{2}}{i^{\alpha}}</math>
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| for some ''α'' > 5 ⁄ 2, then
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| :<math>\frac1{\mu \big( B_{r} (x) \big)} \int_{B_{r} (x)} f(y) \, \mathrm{d} \mu(y) \xrightarrow[r \to 0]{} f(x),</math>
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| for ''γ''-almost all ''x'' and all ''f'' ∈ ''L''<sup>''p''</sup>(''H'', ''γ''; '''R'''), ''p'' > 1.
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| As of 2007, it is still an open question whether there exists an infinite-dimensional Gaussian measure ''γ'' on a separable Hilbert space ''H'' so that, for all ''f'' ∈ ''L''<sup>1</sup>(''H'', ''γ''; '''R'''),
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| :<math>\lim_{r \to 0} \frac1{\gamma \big( B_{r} (x) \big)} \int_{B_{r} (x)} f(y) \, \mathrm{d} \gamma(y) = f(x)</math>
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| for ''γ''-almost all ''x'' ∈ ''H''. However, it is conjectured that no such measure exists, since the ''σ''<sub>''i''</sub> would have to decay very rapidly.
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| ==See also==
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| * [[Differentiation under the integral sign]]
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| ==References==
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| * {{cite book
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| | last = Preiss
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| | first = David
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| | coauthors = Tišer, Jaroslav
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| | chapter = Differentiation of measures on Hilbert spaces
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| | title = Measure theory, Oberwolfach 1981 (Oberwolfach, 1981)
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| | series = Lecture Notes in Math.
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| | volume = 945
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| | pages = 194–207
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| | publisher = Springer
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| | location = Berlin
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| | year = 1982
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| }} {{MathSciNet|id=675283}}
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| * {{cite journal
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| | last = Tišer
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| | first = Jaroslav
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| | title = Differentiation theorem for Gaussian measures on Hilbert space
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| | journal = Trans. Amer. Math. Soc.
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| | volume = 308
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| | year = 1988
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| | issue = 2
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| | pages = 655–666
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| | doi = 10.2307/2001096
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| | publisher = Transactions of the American Mathematical Society, Vol. 308, No. 2
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| | jstor = 2001096
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| }} {{MathSciNet|id=951621}}
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| [[Category:Theorems in analysis]]
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| [[Category:Measure theory]]
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