Page 285 - DMTH401_REAL ANALYSIS

Unit 23: Differentiation of Integrals

Notes
1
lim ò f(y)d n (y) = f(x)
r  n (B (x)) r B (x)
0
r
for  -almost all points x  R . It is important to note, however, that the measure zero set of
n
n
“bad” points depends on the function f.
Borel Measures on R n
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  is any locally finite Borel measure on R and
n
f : R  R is locally integrable with respect to , then
n
1

lim ò f(y)d (y) = f(x)
r  (B (x)) r B (x)
0
r
n
for -almost all points x R .
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 . 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  on a separable Hilbert space H and a Borel set M  H so that,
for -almost all x  H,

 (M B (x))
Ç
lim r = 1
r 0  (B (x))
r
1
 There is a Gaussian measure  on a separable Hilbert space H and a function f  L (H, ; R)
such that
ì 1 ü


r - +¥
s
lim inf í ò Ds(x) f(y)d (y) x II, 0 < < ý
r 0  (B (x))
ï î s ï þ
However, there is some hope if one has good control over the covariance of . Let the covariance
operator of  be S : H  H given by


 Sx, y = H ò  x, z y, z d (z)


or, for some countable orthonormal basis (e ) of H,
i iN
2

Sx = å s  x, e e .
i
1
i
i N

In 1981, Preiss and Jaroslav Tišer showed that if there exists a constant 0 < q < 1 such that
s 2 i 1 £ qs 2 i ,
+
then, for all f  L (H, ; R),
1
1 

ò f(y)d (y) ¾¾¾ f(x)
 (B (x)) r B (x) r 0
r
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