DEMONSTRATIO MATHEMATICA
Vol. XLIII
No 1
2010
Artur Bartoszewicz, Małgorzata Filipczak, Tadeusz Poreda
ON DENSITY WITH RESPECT TO EQUIVALENT MEASURES
Abstract. In the paper there is disscussed a notion of a density point of a Borel
subset of a metric space with respect to a Borel measure µ. There are considered densities
with respect to equivalent measures and density with respect to the limit of a sequence of
equivalent measures.
Let us start from the classical definition of a density of a measurable
subset of the real line at a point x0 ∈ R with respect to Lebesgue measure
λ. A number d ∈ [0, 1] is called the density of a measurable set E ⊂ R at a
point x0 if
λ (E ∩ [x0 − h, x0 + h])
d = lim
.
2h
h→0+
If d = 1 we say that x0 is a density point of a set E.
We shall assume that X is a metric space, µ is a Borel measure on X
and x0 ∈ X belongs to the support of µ i.e. for any ball Br with the center
x0 and arbitrary radius r, we have µ (Br ) > 0. Moreover, let µ be finite
on some ball Br0 . The simple generalization of a notion of the density with
respect to Lebesgue measure leads us to the following
Definition 1. A number d ∈ [0, 1] is called density of a Borel set A at a
point x0 with respect to the measure µ if
µ (A ∩ Br )
d = lim
.
+
µ (Br )
r→0
If d = 1 we say that x0 is a µ-density point of a set A.
Recall that a measure ν is absolutely continuous with respect to the
measure µ (ν ≪ µ), defined on the same σ-algebra F , if from the fact µ(A) =
0 it follows that ν(A) = 0. We say that measures µ and ν are equivalent if
µ ≪ ν and ν ≪ µ. It is well known, that if µ and ν are σ-finite measures on
2000 Mathematics Subject Classification: Primary 28A05; Secondary 28A25.
Key words and phrases: density points, Borel measures, equivalent measures.
22
A. Bartoszewicz, M. Filipczak, T. Poreda
(X, F ) and ν ≪ µ, then there exists a nonnegative F -measurable function
f (called the Radon-Nikodym derivative of ν withTrespect to µ and denoted
dν
by dµ
) such that for any A ∈ F we have ν(A) = f dµ.
A
Since equivalent measures have the same σ-algebras of measurable sets
and σ-ideals of null sets, there is a natural question of densities generated by
equivalent measures. There exist equivalent measures which generated different densities (such measures shall
T be constructed in the proof of Theorem
1). In [1] it is shown that if ν = f dµ and
(∗)
0 < m 6 f (x) 6 M
µ-almost everywhere on some neighbourhood of x0
then, for every measurable set A,
(∗∗)
x0 is a µ-density point of A
if and only if x0 is a ν-density point of A.
We shall show that conditions (∗) and (∗∗) are not equivalent. We shall
also consider a density with respect to the measure µ which is a limit of a
sequence of equivalent measures.
Theorem 1. Suppose that a Borel probability measure µ on (X, F ) satisfies
the following condition of continuity at x0 ∈ X: µ (Br ) is a continuous
increasing function of a radius r on some interval (0, r0 ] such that
lim µ (Br ) = 0.
r→0+
Then, for any number d ∈ [0, 1), there exists a measure ν equivalent to µ
and a Borel set A such that the density of A at x0 with respect to ν is equal
to d and x0 is a density point of A with respect to µ.
Proof. Fix d ∈ (0, 1). Let {rn } → 0 be a decreasing sequence of numbers
smaller than r0 . Denote: C0 = X r Br1 , Cn = Brn r Brn+1 and cn = µ (Cn )
for n ∈ N. We can choose rn such that c1 > c2 > . . . , c0 > 1 − d and
cn
(1)
lim ∞
= 0.
n→∞ P
ck
k=n
It is easy to see that
∞
∞
P
cn = 1, because the sets C0 , C1 ,. . . are disjoint, the
n=0
union ∪ Cn is equal to X \ {x0 } and - by the condition lim µ (Br ) = 0 n=0
r→0+
we have µ ({x0 }) = 0. From the continuity of µ at x0 it follows
that for any
n ∈ N there is rn ∈ (rn+1 , rn ) such that µ Brn r Brn+1 = 2cnn . Denote by
23
On density with respect to equivalent measures
An = Cn r Brn and an = µ (An ). Since µ (Cn r An ) =
∞
P
ak
k=n
(2)
lim ∞
= 1.
n→∞ P
ck
cn
2n
, we have
k=n
Let A =
∞
[
An . From (1) and (2) we obtain that x0 is a density point of
n=1
the set A with respect to µ. Indeed, for any rn+1 6 r 6 rn ,
P
∞
∞
P
ak
ak − an
µ A ∩ Brn+1
µ (A ∩ Br )
k=n+1
1>
= k=n ∞
>
= ∞
P
P
µ (Br )
µ (Brn )
ck
ck
k=n
=
∞
P
k=n
∞
P
ak
ck
k=n
−
k=n
an
.
∞
P
ck
k=n
Let us define a function f : X → (0, ∞) by the formula
 cn

d − an

for t ∈ Cn r An , n = 1, 2, . . .


 cn − an
f (t) = 1
for t ∈ An , n = 1, 2, . . .


d
+
c
−
1

0


for t ∈ C0 .
dc0
T
It is easy to see that ν (A) = f dµ is a probability measure equivalent to µ.
A
Moreover, for any n ∈ N, ν (An ) = an and ν (Cn ) = ν (An ) − ν (Cn r An )
= cdn .
Finally, from (1) and (2) we obtain that density of A at x0 with respect
to v is equal to d. Indeed, for any rn+1 < r 6 rn
P
∞
ak − an
ν A ∩ Brn+1
ν (A ∩ Br )
>
= k=n ∞
→d
ν (Br )
ν (Brn )
1 P
ck
d
k=n
and
ν (A ∩ Brn )
ν (A ∩ Br )
=
6
ν (Br )
ν Brn+1
P
∞
ak + an
k=n+1
∞
1 P
ck
d
k=n+1
→ d.
24
A. Bartoszewicz, M. Filipczak, T. Poreda
1
for any n ∈ N. Obviously
Suppose now that d = 0. Let cn = n1 − n+1
{c
}
satisfies
(1).
Changing
the
function
f
on
sets
Cn \An such that f (t) =
n
√
T
√
ncn −an
ncn . By (1), there
cn −an and putting ν(A) = f dµ we obtain ν(Cn ) =
is n0 such that cn <
∞
X
A
ck for n > n0 , and for any r ∈ (rn+1 , rn ] we have
k=n+1
ν(A ∩ Br )
ν(A ∩ Brn )
6
=
ν(Br )
ν(Brn+1 )
∞
P
ak + an
k=n+1
∞
P
∞
P
ck + cn
2
<√
<p
.
∞
√
P
n+1
ck
kck
(n + 1)
k=n+1
k=n+1
k=n+1
Hence
ν(A∩Br )
ν(Br )
tends to 0 when r → 0+ . Moreover,
∞ √
P
ncn < ∞ and
n=1
ν ({x0 }) = 0. We can also change the measure ν on finite many sets Cn such
that kνk = 1.
Remark 1. Observe that our condition of continuity is satisfied at every
point for any measure µ positive on balls and vanishing on spheres.
Indeed, let us consider an arbitrary point x ∈ X and the function
b : (0, ∞) → R defined by the formula b (r) = µ (Br ) = µ (B (x, r)). Of
course b is increasing. Moreover, discontinuity of b in some point r0 means
that the measure µ of the sphere with the origin x and radius r0 is positive.
Finally, observe that x is not an atom because x belongs to some sphere
(with an other origin). Hence lim b (r) = µ ({x}) = 0.
r−→0+
Clearly, functions constructed in the previous proof do not satisfy the
condition (∗) - and measures do not satisfy the condition (**). Using the
sameTset A, we shall construct a function f which does not satisfy (*) but
ν = f dµ satisfies (∗∗). We start from an easy observation.
Remark 2. Suppose that µ is a Borel measure on (X, F ). If densities of
sets A and B at a point x0 with respect to measure µ are equal to zero,
then density of A ∪ B at x0 is zero. If x0 is a µ-density point of sets A and
B then x0 is a µ-density point of A ∩ B.
Theorem 2. For any Borel probability measure µ on (X, F ) and any point
x0 such that µ satisfies the condition of continuity at x0 , there exists an
equivalent Borel measure ν such that x0 is a µ-density point of a set S if and
only if x0 is a ν-density point of S, and for any neighbourhood U of x0 and
number M > 0
µ ({x ∈ U : f (x) > M }) > 0,
where f is a Radon-Nikodym derivative
dν
dµ .
25
On density with respect to equivalent measures
Proof. Let rn , Cn , cn , An , an and A be defined as in the proof of Theorem 1.
Put
(
n for t ∈ Cn \An
f (t) =
1 for other t
T
and ν = f dµ. Obviously, for any Borel set S ⊂ X, ν (S) > µ (S) and
ν (A ∩ S) = µ (A ∩ S). Moreover
cn
cn
n−1
ν (Cn ) = an + n (cn − an ) = cn − n + n n = cn 1 + n
.
2
2
2
So f is integrable, and the measures µ and ν are in fact equivalent.
Observe that we can make ν a probability measure changing values of f on
C0 , C1 , . . . , Ck for some number k. We will prove that x0 is a ν-density
point of A.
For any α > 1 there is n0 ∈ N such that 1 + n20n−1
< α. For r ∈
0
(rn+1 , rn ] with n > n0 we have
∞
∞
P
P
ak
ak
ν A ∩ Brn+1
1 k=n+1
ν (A ∩ Br )
k=n+1
.
>
= ∞
∞
>α P
P
ν (Br )
ν (Brn )
ck 1 + k−1
c
k
2k
k=n
From (2) and (3) we conclude that the last expression tends
lim inf
r→0+
Therefore
k=n
to α1
and
ν (A ∩ Br )
1
> .
ν (Br )
α
ν (A ∩ Br )
= 1.
ν (Br )
Suppose now that x0 is a ν-density point of a Borel set S. For any r > 0
µ (S ∩ Br )
µ (A ∩ S ∩ Br )
ν (A ∩ S ∩ Br )
ν (A ∩ S ∩ Br )
>
=
>
.
µ (Br )
µ (Br )
µ (Br )
ν (Br )
According to Remark 2, x0 is a ν-density point of A ∩ S and consequently
µ (S ∩ Br )
= 1.
lim
r→0+ µ (Br )
Finally, suppose that x0 is a µ-density point of a Borel set S. This time
we shall observe density of a complement S ′ of a set S. For any r > 0
ν (S ′ ∩ Br )
µ (A ∩ S ′ ∩ Br ) ν (A′ ∩ S ′ ∩ Br )
=
+
ν (Br )
ν (Br )
ν (Br )
′
′
µ (S ∩ Br ) ν (A ∩ Br )
6
+
.
µ (Br )
ν (Br )
lim
r→0+
26
A. Bartoszewicz, M. Filipczak, T. Poreda
The first summand tends to zero because x0 is a µ-density point of S, the
second one- because x0 is a ν-density point of A.
In the next part we shall discusse density with recpect to the limit of a
sequence of equivalent measures. We shall use the set A and the function f
constructed in the proof of Theorem 1, and the following result of Kakutani:
Lemma 1. (compare [2] and [3]) Let P , Q, M be probability measures on
(X, F ) such that P ≪ M and Q ≪ M . Denote by p and q Radon-Nikodym
derivatives of P and Q with respect to M . Then
2 sup |P (B) − Q (B)| =
B∈F
\
X
|p − q| dM .
Observe, that for any sequence of measures {µn } there is a measure
∞
P
µn
M=
2n such that every µn is absolutely continuous with respect to M .
n=1
Hence the uniform convergence of probability measures can be described as
a convergence in some L1 space.
Theorem 3. Suppose that µ is a Borel probability measure on (X, F ). For
any d ∈ [0, 1) there exist a Borel set A and a sequence of probability measures
µn equivalent to µ such that
(i) density of A at x0 with respect to every µn is equal to d,
(ii) {µn } tends uniformly to µ,
(iii) x0 is a density point of A with respect to µ.
Proof. Fix d ∈ (0, 1). Let rn , Cn , cn , An , an , A and the function f be
defined as in the proof of Theorem 1. Define fn as follows:
 ck

d − ak

for t ∈ Ck r Ak , k > n



 ck − ak P
∞

1
ck
fn (t) =
.
d −1

k=n

for
t
∈
C
1
−

0

c0



1
for other t
It can be easily verified that every fn is the Radon-Nikodym derivative of
some probability measure µn on X, equivalent to µ. Since fn (t) = f (t) for
∞
[
t∈
Ck , the densities of A at x0 with respect to µn are all equal to d as
k=n
was shown in the proof of Theorem 1. Observe that
∞
∞
∞
X
\
ck X
1 X |fn − f | dµ = −
ak +
ck → 0.
n→∞
d
d
X
k=n
k=n
k=n
On density with respect to equivalent measures
27
If d = 0 we use in the definition of fn the function from the second part
of the proof of Theorem 1 and change it again on finite many sets Cn to
obtain probability measures µn . Again we obtain that {fn } converges to 1
in L1 (µ).
Finally, the sequence {µn } tends uniformly to µ.
At the end we shall consider another kind of convergence of measures,
which is strictly connected with density of a set at a point.
Theorem 4. Let µ and
Assume that densities dn
tend to d and that µµn is
positive measure. Then d
the measure µ.
µn for n = 1, 2, . . . be probability measures on X.
of a Borel set A at a point x0 with respect to µn
convergent to 1 uniformly on all Borel sets with
is the density of A at the point x0 with respect to
Proof. Consider the difference
µ (A ∩ Br )
µ (A ∩ Br ) µn (A ∩ Br ) µn (A ∩ Br )
µ (Br ) − d 6 µ (Br ) − µn (Br ) + µn (Br ) − dn + |d − dn |
= R1 + R2 + R3 .
Fix any ε > 0. For any n ∈ N
µ (A ∩ Br ) µn (Br ) − µn (Br ) µn (A ∩ Br )
R1 = µ (Br ) µn (Br )
µn (Br ) µn (A ∩ Br ) − µ (Br ) µn (A ∩ Br ) +
µ (Br ) µn (Br )
|µ (A ∩ Br ) − µn (A ∩ Br )| |µn (Br ) − µ (Br )|
6
+
µ (Br ∩ A)
µ (Br )
µn (A ∩ Br ) µn (Br ) 6 1 −
+ 1−
,
µ (Br ∩ A) µ (Br ) if only µ (Br ∩ A) > 0. There is n1 , such that R1 < ε for n > n1 , and,
R3 < ε for n greater than some n2 . Finally, we can fix n > max (n1 , n2 ) and
find r0 small enough to have R2 < ε for r < r0 .
If µ (Br ∩ A) = 0 for some r > 0, we can repeat our cosiderations for the
set X \ A.
Remark 3. Consider the sequence {µn } of measures given by RadonNikodym derivatives fn (with respect to some measure M ). Assume that
fn tend almost everywhere uniformly to the function f > c > 0, which is
a Radon-Nikodym derivative of some measure µ. The measures µ, µn are
the natural example of measures satisfying the assumptions of Theorem 4.
28
A. Bartoszewicz, M. Filipczak, T. Poreda
Indeed, for any Borel set B of positive measure we have
T
T
fn dM
|f − f | dM
B n
B
µn (B)
M (B) ess sup |fn − f |
T
6
µ (B) − 1 = Tf dM − 1 6
f dM
c · M (B)
B
B
where ess sup denotes M -essential supremum.
Unfortunately this example is rather trivial, because for almost all x ∈ X
and n large enough
c
c
f (x) − < fn (x) < f (x) + .
2
2
Thus
dµn
fn
dµn
= dM
=
dµ
dµ
f
dM
satisfies (∗), so µn and µ generate the same density.
References
[1] A. Bartoszewicz, On density points of subsets of metric space with respect to the measure given by Radon-Nikodym derivative, Real Anal. Exchange 23(2) (1997/8), 783–
786.
[2] S. Kakutani, On equivalence of infinite product measures, Ann. Math. 49 (1948), 214–
224.
[3] C. Kraft, Some conditions for consistency and uniform consistency of statistical procedures, Univ. of Calif. Publ. Stat. 2.6 (1955), 125–141.
[4] J. C. Oxtoby, Measure and Category, Springer, New York, 1971.
Artur Bartoszewicz, Tadeusz Poreda
INSTITUTE OF MATHEMATICS, ŁÓDŹ TECHNICAL UNIVERSITY
ul. Wólczańska 215
93-005 ŁÓDŹ, POLAND
E-mail: [email protected]
[email protected]
Małgorzata Filipczak
FACULTY OF MATHEMATICS AND COMPUTER SCIENCES
ŁÓDŹ UNIVERSITY
ul. S. Banacha 22
90-238 ŁÓDŹ, POLAND
E-mail: malfi[email protected]
Received November 4, 2008; revised version July 4, 2009.