3. Null-Continuity and Convergence of a Sequence of Measurable Functions
Let be a monotone measure space and be the class of all finite real-valued -measurable functions on and let denote the class of all finite nonnegative real-valued -measurable functions on .
Let (). We say that converges almost everywhere to f on X , and denote it by , if there is a subset such that and on ; converges almost uniformly to f on X, and denote it by , if, for any , there is a subset such that and converges to f uniformly on ; converge to f in measure μ on X, in symbols , if, for any , .
Let . We say a proposition P with respect to points in A is almost everywhere true on A, denoted by “ on A”, if there exists such that and P is true on . When , “ on A" is denoted by “”. For example, for , , means that .
In [
1], the relations among the null-continuity and other different continuity were discussed. Now, we use the convergence of sequence of measurable functions to describe the null-continuity.
Proposition 5. Let be a monotone measure space. Then, the following statements are equivalent:
- (1)
μ is null-continuous;
- (2)
For any , with , if , then ;
- (3)
For any , with , if , then , where C is a constant.
Proof. (1) ⇒ (2). Suppose that is null-continuous. For and , denote and ). If , and , then and . Noting that and is null-continuous, we have , i.e., .
(2) ⇒ (1). For every increasing sequence such that , it is easy to see that (where , and are the characteristic functions of and A, respectively) and . It follows from condition (2) that , i.e., . Therefore, is null-continuous.
(1) ⇔ (3). The proof is similar. □
The Riesz theorem is one of the most important convergence theorems in classical measure theory. It states that, to each sequence of measurable functions which converges in measure, there is a subsequence converging almost everywhere, i.e., for any
and
, if
, then there exists a subsequence
of
such that
(see [
10]).
This important theorem was generalized to monotone measure spaces. In [
8], the concept of property
of monotone measures was introduced, and it was shown that the conclusion of the classical Riesz theorem holds for a monotone measure
if and only if
has property
([
8], also see [
5,
7,
21]). We recall these results.
Definition 4 ([
8])
. A monotone measure μ is called to have property , if for any with , there exists a subsequence of such that . The following result is a version of Riesz’s theorem for monotone measures [
8]:
Theorem 1. Let be a monotone measure space. Then, the following statements are equivalent:
- (1)
μ has property .
- (2)
For any and , if , then there exists a subsequence of such that .
The following implication between the property (S) and null-continuity is shown in [
6]:
Proposition 6. Let be a monotone measure space. If μ has property (S), then μ is null-continuous.
Thus, we obtain a necessary condition that Riesz’s theorem holds for monotone measures:
Proposition 7. Let be a monotone measure space. If the classical Riesz’s theorem remains valid for the monotone measure μ, then μ is null-continuous.
Note that null-continuity may not imply property (S) (see [
6]).
In the following, we concentrate on the discussion of convergence of measurable functions on S-compact spaces.
A measurable space
is said to be
S-compact, if, for any sequence of sets in
, there exists some convergent subsequence, i.e.,
,
such that
Observe that, if
X is countable, then
is
S-compact space ([
7]). The converse may not be true (see [
7]).
When is an S-compact space, the converse of Proposition 6 is true:
Proposition 8. Let be an S-compact space and . Then, μ is null-continuous if and only if μ has property (S).
Proof. If
has property
, it follows from Proposition 6 that
is null-continuous. Conversely, assume
is null-continuous. For any
with
, since
is
S-compact, so there exists a subsequence
of
such that
For any fixed
, since
and
it follows that
. By the null-continuous and Equation (
2), then
. Therefore,
has property (S). □
The following is a direct consequence of Propositions 2 and 8:
Corollary 1. Let be an S-compact space and . If μ is strongly order continuous and weakly null-additive, then μ has property (S).
Combining Theorem 1 and Proposition 6, we obtain a version of Riesz’s theorem for monotone measure on S-compact spaces.
Theorem 2. (Riesz’s theorem) Let be an S-compact space and . Then, the following statements are equivalent:
- (1)
μ is null-continuous.
- (2)
For any and , if , then there exists a subsequence of such that .
From Propositions 2, 4, and Theorem 2, we obtain the following corollaries.
Corollary 2. Let be an S-compact space and let be strongly order continuous and weakly null-additive. If , then there exists a subsequence of such that .
Corollary 3. Let be an S-compact space and . Then, each of the conditions (i)–(v) in Proposition 4 is a sufficient condition that the conclusion of the classical Riesz theorem holds for monotone measure μ.
Egoroff’s theorem is one of the most important convergence theorems in classical measure theory. It states that almost everywhere convergence implies almost uniform convergence on a measurable set of finite measure ([
17,
22]). In general, the Egoroff theorem is not valid for monotone measure without additional conditions. This well-known theorem was generalized to monotone measure spaces ([
23,
24]).
By using the equivalence between property and null-continuity on S-compact spaces, we present a version of Egoroff’s theorem for monotone measures on S-compact spaces, as follows:
Theorem 3 (Egoroff’s theorem)
. Let be an S-compact space and let be weakly null-additive and strongly order continuous. Then, for any and , Proof. From Propositions 2 and 8, then
has property
. By Egoroff’s theorem for monotone measures (see Theorem 1 in [
23]), we get the conclusion (
3). □
Observe that weak null-additivity and strong order continuity are independent of each other, as shown in Examples 3 and 4.
By using the fact that null-additivity implies property (S) on
S-compact spaces (Proposition 8), we can obtain the following results (see also [
25]).
Theorem 4. Let be an S-compact space and be strongly order continuous. The following statements are equivalent:
- (1)
μ is weakly null-additive;
- (2)
μ is weakly asymptotic null-additive;
- (3)
μ has (p.g.p.);
- (4)
For any and , if and , then ;
- (5)
For any and , if and , then .
Proof. It is obvious that
. Now, we prove
. Suppose that
is weakly null-additive. Since
is strongly order continuous, from Proposition 2, then
is null-continuous and hence it follows from Proposition 8 that
has property (S). We assume that
has not (p.g.p.). Then, there exist
and two sequences
and
such that
By using property (S), there exist subsequences
and
such that
By the weak null-additivity of
, we have
Therefore, from the strong order continuity of
, we have
This is in contradiction with the fact that
This shows that
. Therefore,
.
For other equivalences, see [
25]. □
The following is a generalization of result in classical measure theory (see [
22]).
Theorem 5. Let be an S-compact space and be weakly null-additive and strongly order continuous. If and , then
Proof. For any given
, we have
Since
and
, we have
From Theorem 4, we know that
has
, and therefore
Hence, it is clear that
Denote
, then
is an increasing sequence of measurable subsets in
and
. Noting that, from Proposition 2,
is null-continuous, then
i.e.,
. □
Similarly, we have the following:
Theorem 6. Let be an S-compact space and be weakly null-additive and strongly order continuous. If and such that , then .
4. Null-Continuity and Integrals
We recall two kinds of basic nonlinear integrals based on monotone measures, namely, the Sugeno integral (also called fuzzy integral) [
15] and the Choquet integral [
14].
Let be a monotone measure space and let be fixed, and .
The Sugeno integral (or fuzzy integral) of
f on
X with respect to
, is defined by
The Choquet integral of
f on
X with respect to
, is defined by
where the right-hand side integral is the improper Riemann integral.
For , define and .
Note that when is a -additive measure, the Choquet integral coincides with the Lebesgue integral. Thus, it is a generalization of the Lebesgue integral. The Sugeno integral, which is based on monotone measure and relates the logic addition and logic multiplication , is a special kind of nonlinear integral. It is not linear even for a probability measure, so the Sugeno integral is not a generalization of the Lebesgue integral.
We recall some basic properties of the Sugeno integral and the Choquet integral, as follows:
Proposition 9. For any , , we have
- (i)
if then ;
- (ii)
if on A, then ;
- (iii)
if , then ,
where the integral in the above formulas stands for the Sugeno integral or the Choquet integral.
Proposition 10 ([
7])
. Let , , . Then,(i) ; .
(ii) , in particular,
Proposition 11. Let be a monotone measure space, and . If on A, then and .
Theorem 7. Let be a monotone measure space and μ be null-continuous. For any and , if , then on A.
Proof. We use a proof by contradiction. Assume the conclusion is not true, i.e.,
Denote
, then
Note that
is an increasing sequence of measurable subsets in
and
is null-continuous, so there exists
such that
(otherwise, by the null-continuity of
,
. Consequently, we have
This contradicts
. □
Theorem 8. Let be a monotone measure space and μ be null-continuous. For any and , if , then on A.
Proof. Denote
. From Propositions 9 and 10, for any
we have
If
, then for any
,
. Noting that
and
, by the null-continuity of
, we have
, i.e.,
on
A. □
As special results of Theorems 7 and 8, we have the following:
Corollary 4 ([
7])
. Let be a monotone measure space. If μ is continuous from below, then the conclusions of Theorem 7 and 8 hold. Corollary 5. Let be a monotone measure space. If μ satisfies one of the following conditions:
- (i)
μ is countably weakly null-additive;
- (ii)
μ is order continuous and null-additive;
- (iii)
μ is order continuous and sub-additive;
- (iv)
μ is order continuous and weakly asymptotic null-additive;
- (v)
μ is σ-subadditive,
then the conclusions of Theorems 7 and 8 hold.
Proposition 12. Let be a monotone measure space. Then, the following statements are equivalent:
- (1)
μ is null-continuous;
- (2)
For any , with , if , then - (3)
For any , with , if , then
Proof. We only prove (1) ⇔ (2), for (1) ⇔ (3) the proof is similar. If is null-continuous, then, from Proposition 5, we have . Therefore, it follows from Proposition 11 that . Conversely, if condition (2) holds, then it follows from Theorem 7 that implies . Using Proposition 5 again, then is null-continuous. □
In the following, we discuss the null-continuity of monotone measures defined by integral.
Given
and
, then the Choquet integral (resp. the Sugeno integral) of
f with respect to
determines a new monotone measure
(resp.
), as follows:
Such the new monotone measures
and
are absolutely continuous with respect to the monotone measure
, respectively, i.e.,
and
(see [
7,
17]).
In the following, we show that the new monotone measures and preserve the structural characteristic of null-continuity of the original monotone measure .
Theorem 9. Let be a monotone measure space. If μ is null-continuous, then so are both and .
Proof. We only prove the case of ; for , the proof is similar.
Let
be an increasing sequence with
. Then,
and, from Theorem 8, we have
on
, i.e.,
. Since
and
is null-continuous, then we have
and hence
on
. Therefore,
, i.e.,
. This shows that
is null-continuous. □
5. Pan-Integral and Concave Integral
In the above discussions, we only discussed the Sugeno integral and the Choquet integral. There are two other important nonlinear integrals, the pan-integral [
7,
11], and the concave integral [
12,
13] (see also [
26,
27]), as follows:
Given and .
The
pan-integral of
f on
X with respect to
(and the standard addition + and multiplication ·) is defined by
where
are the characteristic functions of
, respectively.
The
concave integral of
f on
X is defined by
where
are the characteristic functions of
, respectively.
Note that, when is a -additive measure, the Choquet integral, the pan-integral, and the concave integral coincide with the Lebesgue integral. Thus, these three integrals are all generalizations of the Lebesgue integral. However, they are significantly different from each other. Observing that the Choquet integral is based on chains of measurable subsets of X, the pan-integral is related to finite measurable partitions of X and the concave integral to any finite set systems of measurable subsets of X.
In general, for any
, we have
but the converse inequalities may not be true, and
and
are incomparable (see [
28,
29,
30]).
For the pan-integral and the concave integral, the conclusions of Theorem 7 (or Theorem 8) and Proposition 12 remain valid.
Theorem 10. Let be a monotone measure space and μ be null-continuous. For any and , if (or ), then on A.
Proof. It is similar to the proof of Theorem 8. □
Proposition 13. Let be a monotone measure space. Then, the following statements are equivalent:
- (1)
μ is null-continuous;
- (2)
For any , with , if , then - (3)
For any , with , if , then
Proof. It is similar to the proof of Proposition 12. □