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 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 for λn-almost all points x∈Rn. 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 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

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 or, for some countable orthonormal basis (ei)i∈N of H

In 1981, Preiss and Jaroslav Tišer showed that if there exists a constant 0q then, for all f∈L1(H,γ;R where the convergence is convergence in measure with respect to γ for some α5⁄ for γ-almost all x and all f∈Lp(H,γ;R), p1.

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