Differentiation of integrals

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 (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),

\( {\frac 1{\mu {\big (}B_{{r}}(x){\big )}}}\int _{{B_{{r}}(x)}}f(y)\,{\mathrm {d}}\mu (y){\xrightarrow[ {r\to 0}]{\gamma }}f(x), \)

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 ∈ 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.