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

\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

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}}{\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

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

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

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 675283.
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 951621.

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