View source for Differentiation of integrals

{{Short description|Problem in mathematics}}
In [[Mathematics|mathematics]], the problem of '''differentiation of integrals''' is that of determining under what circumstances the [[Average|mean value]] [[Integral|integral]] of a suitable [[Function (mathematics)|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 (mathematics)|measure]] ''μ'' and a [[Metric space|metric]] ''d'', one asks for what functions ''f''&nbsp;:&nbsp;''X''&nbsp;→&nbsp;'''R''' does

:<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>

for all (or at least ''μ''-[[Almost all|almost all]]) ''x''&nbsp;∈&nbsp;''X''?  (Here, as in the rest of the article, ''B''<sub>''r''</sub>(''x'') denotes the open ball in ''X'' with ''d''-[[Radius|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''.

== Theorems on the differentiation of integrals ==

=== Lebesgue measure ===

One result on the differentiation of integrals is the [[Lebesgue differentiation theorem]], as proved by [[Biography:Henri Lebesgue|Henri Lebesgue]] in 1910.  Consider ''n''-[[Physics:Dimension|dimension]]al [[Lebesgue measure]] ''λ''<sup>''n''</sup> on ''n''-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup>.  Then, for any [[Locally integrable function|locally integrable function]] ''f''&nbsp;:&nbsp;'''R'''<sup>''n''</sup>&nbsp;→&nbsp;'''R''', one has

:<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>

for ''λ''<sup>''n''</sup>-almost all points ''x''&nbsp;∈&nbsp;'''R'''<sup>''n''</sup>.  It is important to note, however, that the measure zero set of "bad" points depends on the function ''f''.

=== Borel measures on R<sup>''n''</sup> ===

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''&nbsp;:&nbsp;'''R'''<sup>''n''</sup>&nbsp;→&nbsp;'''R''' is locally integrable with respect to ''μ'', then

:<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>

for ''μ''-almost all points ''x''&nbsp;∈&nbsp;'''R'''<sup>''n''</sup>.

=== Gaussian measures ===

The problem of the differentiation of integrals is much harder in an infinite-dimensional setting.  Consider a [[Separable space|separable]] [[Hilbert space]] (''H'',&nbsp;⟨&nbsp;,&nbsp;⟩) 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:
* There is a Gaussian measure ''γ'' on a separable Hilbert space ''H'' and a Borel set ''M''&nbsp;⊆&nbsp;''H'' so that, for ''γ''-almost all ''x''&nbsp;∈&nbsp;''H'', <math display="block">\lim_{r \to 0} \frac{\gamma \big( M \cap B_{r} (x) \big)}{\gamma \big( B_{r} (x) \big)} = 1.</math>
* There is a Gaussian measure ''γ'' on a separable Hilbert space ''H'' and a function ''f''&nbsp;∈&nbsp;''L''<sup>1</sup>(''H'',&nbsp;''γ'';&nbsp;'''R''') such that <math display="block">\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>

However, there is some hope if one has good control over the [[Covariance|covariance]] of ''γ''. Let the covariance operator of ''γ'' be ''S''&nbsp;:&nbsp;''H''&nbsp;→&nbsp;''H'' given by

:<math>\langle Sx, y \rangle = \int_{H} \langle x, z \rangle \langle y, z \rangle \, \mathrm{d} \gamma(z),</math>

or, for some [[Countable set|countable]] [[Orthonormal basis|orthonormal basis]] (''e''<sub>''i''</sub>)<sub>''i''∈'''N'''</sub> of ''H'',

:<math>Sx = \sum_{i \in \mathbf{N}} \sigma_{i}^{2} \langle x, e_{i} \rangle e_{i}.</math>

In 1981, Preiss and Jaroslav Tišer showed that if there exists a constant 0&nbsp;&lt;&nbsp;''q''&nbsp;&lt;&nbsp;1 such that

:<math>\sigma_{i + 1}^{2} \leq q \sigma_{i}^{2},</math>

then, for all ''f''&nbsp;∈&nbsp;''L''<sup>1</sup>(''H'',&nbsp;''γ'';&nbsp;'''R'''),

:<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>

where the convergence is [[Convergence in measure|convergence in measure]] with respect to ''γ''.  In 1988, Tišer showed that if

:<math>\sigma_{i + 1}^{2} \leq \frac{\sigma_{i}^{2}}{i^{\alpha}}</math>

for some ''α''&nbsp;&gt;&nbsp;5&nbsp;⁄&nbsp;2, then

:<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>

for ''γ''-almost all ''x'' and all ''f''&nbsp;∈&nbsp;''L''<sup>''p''</sup>(''H'',&nbsp;''γ'';&nbsp;'''R'''), ''p''&nbsp;&gt;&nbsp;1.

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''&nbsp;∈&nbsp;''L''<sup>1</sup>(''H'',&nbsp;''γ'';&nbsp;'''R'''),

:<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>

for ''γ''-almost all ''x''&nbsp;∈&nbsp;''H''.  However, it is conjectured that no such measure exists, since the ''σ''<sub>''i''</sub> would have to decay very rapidly.

==See also==

* {{annotated link|Differentiation rules}}
* {{annotated link|Leibniz integral rule}}
* {{annotated link|Reynolds transport theorem}}

== References ==
* {{cite book
| last1 = Preiss
| first1 = David
|last2=Tišer|first2= Jaroslav
| chapter = Differentiation of measures on Hilbert spaces
| title = Measure theory, Oberwolfach 1981 (Oberwolfach, 1981)
| series = Lecture Notes in Mathematics
| volume = 945
| pages = 194–207
| publisher = Springer
| location = Berlin
| year = 1982
| mr = 675283
|doi=10.1007/BFb0096675
| isbn = 978-3-540-11580-9
}}
* {{cite journal
| last = Tišer
| first = Jaroslav
| title = Differentiation theorem for Gaussian measures on Hilbert space
| journal = Transactions of the American Mathematical Society
| volume = 308
| year = 1988
| issue = 2
| pages = 655–666
| doi = 10.2307/2001096
| jstor = 2001096
|url=https://www.ams.org/journals/tran/1988-308-02/S0002-9947-1988-0951621-8/S0002-9947-1988-0951621-8.pdf
| mr = 951621}}

{{Calculus topics}}

[[Category:Differentiation rules]]
[[Category:Measure theory]]
[[Category:Theorems in analysis]]
[[Category:Theorems in calculus]]

{{Sourceattribution|Differentiation of integrals}}