Differentiation of integrals


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 x ∈ 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 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 R''n''

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 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 show the kind of difficulties that one can expect to encounter in this setting:
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 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,
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, 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,
for γ-almost all x ∈ H. However, it is conjectured that no such measure exists, since the σi would have to decay very rapidly.