Differentiation_of_integrals Knowpia

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

\lim _{r\to 0}{\frac {1}{\mu {\big (}B_{r}(x){\big )}}}\int _{B_{r}(x)}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 edit

Lebesgue measure edit

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}{\frac {1}{\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 edit

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}{\frac {1}{\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 edit

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

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

\langle Sx,y\rangle =\int _{H}\langle x,z\rangle \langle y,z\rangle \,\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}\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),

{\frac {1}{\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

{\frac {1}{\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 ∈ 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),

\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 σ_i would have to decay very rapidly.

References edit

Preiss, David; Tišer, Jaroslav (1982). "Differentiation of measures on Hilbert spaces". Measure theory, Oberwolfach 1981 (Oberwolfach, 1981). Lecture Notes in Mathematics. Vol. 945. Berlin: Springer. pp. 194–207. doi:10.1007/BFb0096675. ISBN 978-3-540-11580-9. 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.