Differentiation of integrals: Difference between revisions
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* [[Differentiation under the integral sign]] |
* [[Differentiation under the integral sign]] |
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==References== |
== References == |
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* {{cite book |
* {{cite book |
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| last = Preiss |
| last = Preiss |
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| first = David |
| first = David |
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| |
|last2=Tišer|first2= Jaroslav |
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| chapter = Differentiation of measures on Hilbert spaces |
| chapter = Differentiation of measures on Hilbert spaces |
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| title = Measure theory, Oberwolfach 1981 (Oberwolfach, 1981) |
| title = Measure theory, Oberwolfach 1981 (Oberwolfach, 1981) |
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| series = Lecture Notes in |
| series = Lecture Notes in Mathematics |
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| volume = 945 |
| volume = 945 |
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| pages = |
| pages = 194–207 |
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| publisher = Springer |
| publisher = Springer |
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| location = Berlin |
| location = Berlin |
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| year = 1982 |
| year = 1982 |
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| mr = 675283 |
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|doi=10.1007/BFb0096675 |
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}} |
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* {{cite journal |
* {{cite journal |
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| last = Tišer |
| last = Tišer |
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| first = Jaroslav |
| first = Jaroslav |
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| title = Differentiation theorem for Gaussian measures on Hilbert space |
| title = Differentiation theorem for Gaussian measures on Hilbert space |
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| journal = |
| journal = Transactions of the American Mathematical Society |
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| volume = 308 |
| volume = 308 |
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| year = 1988 |
| year = 1988 |
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| issue = 2 |
| issue = 2 |
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| pages = |
| pages = 655–666 |
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| doi = 10.2307/2001096 |
| 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 |
| jstor = 2001096 |
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| mr = 951621}} |
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[[Category:Theorems in analysis]] |
[[Category:Theorems in analysis]] |
Revision as of 20:40, 7 May 2015
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.
Theorems on the differentiation of integrals
Lebesgue measure
One result on the differentiation of integrals is the Lebesgue differentiation theorem, as proved by Henri Lebesgue in 1910. Consider n-dimensional Lebesgue measure λn on n-dimensional Euclidean space Rn. Then, for any locally integrable function f : Rn → R, one has
for λn-almost all points x ∈ Rn. It is important to note, however, that the measure zero set of "bad" points depends on the function f.
Borel measures on Rn
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 Borel measure on Rn and f : Rn → R is locally integrable with respect to μ, then
for μ-almost all points x ∈ Rn.
Gaussian measures
The problem of the differentiation of integrals is much harder in an infinite-dimensional setting. Consider a 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:
- 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,
- There is a Gaussian measure γ on a separable Hilbert space H and a function f ∈ L1(H, γ; R) such that
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
or, for some countable orthonormal basis (ei)i∈N of H,
In 1981, Preiss and Jaroslav Tišer showed that if there exists a constant 0 < q < 1 such that
then, for all f ∈ L1(H, γ; R),
where the convergence is convergence in measure with respect to γ. In 1988, Tišer showed that if
for some α > 5 ⁄ 2, then
for γ-almost all x and all f ∈ Lp(H, γ; R), p > 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 ∈ L1(H, γ; R),
for γ-almost all x ∈ H. However, it is conjectured that no such measure exists, since the σi would have to decay very rapidly.
See also
References
- Preiss, David; Tišer, Jaroslav (1982). "Differentiation of measures on Hilbert spaces". Measure theory, Oberwolfach 1981 (Oberwolfach, 1981). Lecture Notes in Mathematics. Vol. 945. Berlin: Springer. pp. 194–207. doi:10.1007/BFb0096675. MR 0675283.
- Tišer, Jaroslav (1988). "Differentiation theorem for Gaussian measures on Hilbert space". Transactions of the American Mathematical Society. 308 (2): 655–666. doi:10.2307/2001096. JSTOR 2001096. MR 0951621.