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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
Theorems on the differentiation of integrals edit
Lebesgue measure edit
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
Borel measures on Rn edit
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
Gaussian measures edit
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
In 1981, Preiss and Jaroslav Tišer showed that if there exists a constant 0 < q < 1 such that
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),
See also edit
- Differentiation rules – Rules for computing derivatives of functions
- Leibniz integral rule – Differentiation under the integral sign formula
- Reynolds transport theorem – 3D generalization of the Leibniz integral rule
References edit
- 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. ISBN 978-3-540-11580-9. MR 0675283.
- Tišer, Jaroslav (1988). "Differentiation theorem for Gaussian measures on Hilbert space" (PDF). Transactions of the American Mathematical Society. 308 (2): 655–666. doi:10.2307/2001096. JSTOR 2001096. MR 0951621.