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 \lim_{r \to 0} \frac1{\lambda^{n} \big( B_{r} (x) \big)} \int_{B_{r} (x)} 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 \lim_{r \to 0} \frac1{\mu \big( B_{r} (x) \big)} \int_{B_{r} (x)} 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, \lim_{r \to 0} \frac{\gamma \big( M \cap B_{r} (x) \big)}{\gamma \big( B_{r} (x) \big)} = 1. • There is a Gaussian measure
γ on a separable Hilbert space
H and a function
f ∈
L1(
H,
γ;
R) such that \lim_{r \to 0} \inf \left\{ \left. \frac1{\gamma \big( B_{s} (x) \big)} \int_{B_{s} (x)} f(y) \, \mathrm{d} \gamma(y) \right| x \in H, 0 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 \langle Sx, y \rangle = \int_{H} \langle x, z \rangle \langle y, z \rangle \, \mathrm{d} \gamma(z), or, for some
countable orthonormal basis (
ei)
i∈
N of
H, Sx = \sum_{i \in \mathbf{N}} \sigma_{i}^{2} \langle x, e_{i} \rangle e_{i}. In 1981, Preiss and Jaroslav Tišer showed that if there exists a constant 0 <
q < 1 such that \sigma_{i + 1}^{2} \leq q \sigma_{i}^{2}, then, for all
f ∈
L1(
H,
γ;
R), \frac1{\mu \big( B_{r} (x) \big)} \int_{B_{r} (x)} f(y) \, \mathrm{d} \mu(y) \xrightarrow[r \to 0]{\gamma} f(x), where the convergence is
convergence in measure with respect to
γ. In 1988, Tišer showed that if \sigma_{i + 1}^{2} \leq \frac{\sigma_{i}^{2}}{i^{\alpha}} for some
α > 5 ⁄ 2, then \frac1{\mu \big( B_{r} (x) \big)} \int_{B_{r} (x)} f(y) \, \mathrm{d} \mu(y) \xrightarrow[r \to 0]{} f(x), 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), \lim_{r \to 0} \frac{1}{\gamma \big( B_{r} (x) \big)} \int_{B_{r} (x)} 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
σi would have to decay very rapidly. ==See also==