# 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 (or at least *μ*-almost all) *x* ∈ *X*? (Here, as in the rest of the article, *B*_{r}(*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 *λ*^{n} on *n*-dimensional Euclidean space **R**^{n}. Then, for any locally integrable function *f* : **R**^{n} → **R**, one has

for *λ*^{n}-almost all points *x* ∈ **R**^{n}. It is important to note, however, that the measure zero set of "bad" points depends on the function *f*.

### Borel measures on R^{n}

^{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 **R**^{n} and *f* : **R**^{n} → **R** is locally integrable with respect to *μ*, then

for *μ*-almost all points *x* ∈ **R**^{n}.

### 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*∈*L*^{1}(*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 (*e*_{i})_{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* ∈ *L*^{1}(*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* ∈ *L*^{p}(*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* ∈ *L*^{1}(*H*, *γ*; **R**),

for *γ*-almost all *x* ∈ *H*. However, it is conjectured that no such measure exists, since the *σ*_{i} would have to decay very rapidly.

## 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.