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

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(HγR) 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)iN 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(HγR),

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(HγR), 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(HγR),

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. 945. Berlin: Springer. pp. 194–207. doi:10.1007/BFb0096675. 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. 
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