Richardson's theorem

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

$\underset{r\to 0}{\lim }\frac{1}{\mu (B_r(x))}\underset{B_r(x)}{\int }f(y)\mathrm{d}\mu (y)=f(x)$

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 λⁿ on n-dimensional Euclidean space Rⁿ. Then, for any locally integrable function f : Rⁿ → R, one has

$\underset{r\to 0}{\lim }\frac{1}{{\lambda }^n(B_r(x))}\underset{B_r(x)}{\int }f(y)\mathrm{d}{\lambda }^n(y)=f(x)$

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

$\underset{r\to 0}{\lim }\frac{1}{\mu (B_r(x))}\underset{B_r(x)}{\int }f(y)\mathrm{d}\mu (y)=f(x)$

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,

$\underset{r\to 0}{\lim }\frac{\gamma (M\cap B_r(x))}{\gamma (B_r(x))}=1.$

• There is a Gaussian measure γ on a separable Hilbert space H and a function f ∈ L¹(H, γ; R) such that

$\underset{r\to 0}{\lim }\inf \{\frac{1}{\gamma (B_s(x))}\underset{B_s(x)}{\int }f(y)\mathrm{d}\gamma (y)|x\in H,0<s<r\}=+\mathrm{\infty }.$

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

$⟨Sx,y⟩=\underset{H}{\int }⟨x,z⟩⟨y,z⟩\mathrm{d}\gamma (z),$

or, for some countable orthonormal basis (e_i)_i∈N of H,

$Sx=\sum _{i\in \mathbf{N}}{\sigma }_i^2⟨x,e_i⟩e_i.$

In 1981, Preiss and Jaroslav Tišer showed that if there exists a constant 0 < q < 1 such that

${\sigma }_{i+1}^2\le q{\sigma }_i^2,$

then, for all f ∈ L¹(H, γ; R),

$\frac{1}{\mu (B_r(x))}\underset{B_r(x)}{\int }f(y)\mathrm{d}\mu (y)\underset{r\to 0}{\overset{\gamma }{\to }}f(x),$

where the convergence is convergence in measure with respect to γ. In 1988, Tišer showed that if

${\sigma }_{i+1}^2\le \frac{{\sigma }_i^2}{i^{\alpha }}$

for some α > 5 ⁄ 2, then

$\frac{1}{\mu (B_r(x))}\underset{B_r(x)}{\int }f(y)\mathrm{d}\mu (y)\underset{r\to 0}{\overset{}{\to }}f(x),$

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 ∈ L¹(H, γ; R),

$\underset{r\to 0}{\lim }\frac{1}{\gamma (B_r(x))}\underset{B_r(x)}{\int }f(y)\mathrm{d}\gamma (y)=f(x)$

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.

See also

• Differentiation under the integral sign

References

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