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.
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.