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On Lusin's theorem for non-additive measures that take values in an ordered topological vector space
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ScienceDirect Fuzzy Sets and Systems 244 (2014) 41–50 www.elsevier.com/locate/fss
On Lusin’s theorem for non-additive measures that take values in an ordered topological vector space Toshikazu Watanabe ∗ , Tamaki Tanaka Graduate School of Science and Technology, Niigata University, 8050, Ikarashi 2-no-cho, Nishi-ku, Niigata, 950-2181, Japan Received 1 December 2011; received in revised form 16 July 2013; accepted 26 July 2013 Available online 6 August 2013
Abstract Lusin’s theorem was established for real-valued monotone measures under an equivalent condition to Egoroff’s theorem recently. In this paper, we show that the same result remains valid for non-additive measures that take values in an ordered topological vector space. We apply our result to several ordered topological vector spaces. © 2013 Elsevier B.V. All rights reserved. Keywords: Non-additive measure; Egoroff’s theorem; Lusin’s theorem; Ordered vector space; Locally full topology; Pseudometric generating property
1. Introduction The regularity of measure on a topological space is an important property in the measure theory. In fact, Lusin’s theorem can be discussed using such arguments in a metric space. For real-valued fuzzy measure, in [17], Wu and Ha generalized Lusin’s theorem from a classical measure space to a finite autocontinuous fuzzy measure space. Jiang et al. [5,6] extended the result of Wu and Ha. Li and Yasuda showed Lusin’s theorem for real-valued fuzzy measures on a metric space in [11]. Kawabe proved Lusin’s theorem for Riesz space-valued fuzzy Borel measures; see [7]. In [16], the authors also proved Lusin’s theorem for non-additive Borel measures which take values in an ordered topological vector space under the following assumptions; the measure is continuous from above with a property suggested by Sun [13] and weakly null-additive, and the image space has an ordered topological vector space version of the Egoroff property. Recently in [9], Li and Mesiar proved Lusin’s theorem for real-valued monotone measures on a metric space under an equivalent condition to Egoroff’s theorem by using the pseudometric generating property of set function. In this paper, motivated by [9], we prove Lusin’s theorem for non-additive Borel measures which take values in an ordered topological vector space under the measure has the pseudometric generating property and satisfies the Egoroff condition. We apply our result to several ordered topological vector spaces. * Corresponding author.
E-mail addresses: [email protected] (T. Watanabe), [email protected] (T. Tanaka). 0165-0114/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2013.07.021
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2. Preliminaries Let R be the set of real numbers and N the set of natural numbers. Denote by Θ the set of all mappings from N into N . Let X be a non-empty set and F a σ -field of X. A topology on a vector space E is called a vector topology if the mappings (x, y) → x + y and (α, x) → αx are continuous for any x, y ∈ E and α ∈ R. A subset F of an ordered vector space E is called full if x1 , x2 ∈ F and x1 x2 implies [x1 , x2 ] = {x ∈ E | x1 x x2 } ⊂ F . We consider a vector topology on E and let B0 be a system of neighborhoods of 0 ∈ E. The vector topology on E is called locally full, if there exists a basis of B0 consisting of full sets. An ordered vector space endowed with this topology is called an ordered topological vector space; see [3]. We also assume that E is a Hausdorff space with the first axiom of countability. Let {un } be a sequence in E and u ∈ E. We say that un converges to u and write un → u if for any U ∈ B0 , there exists an n0 ∈ N such that un − u ∈ U for any n n0 . Definition 1. A set function μ : F → E is called a non-additive measure if it satisfies the following two conditions: (1) μ(∅) = 0. (2) If A, B ∈ F and A ⊂ B, then μ(A) μ(B). Definition 2. Let μ : F → E be a non-additive measure. μ is said to be continuous from above if μ(An ) → μ(A) whenever {An } ⊂ F and A ∈ F satisfy An A. μ is said to be continuous from below if μ(An ) → μ(A) whenever {An } ⊂ F and A ∈ F satisfy An A. μ is called a fuzzy measure if it is continuous from above and below. μ is said to be strongly order continuous if it is continuous from above at measurable sets of measure 0, that is, for any {An } ⊂ F and A ∈ F with An A and μ(A) = 0, it holds that μ(An ) → 0; see [8]. (5) μ has ∞ (S) if for any sequence {An } ⊂ F with μ(An ) → 0, there exists a subsequence {Ank } such that property μ( ∞ i=1 k=i Ank ) = 0; see [13]. (6) μ is said to be weakly null-additive if μ(A ∪ B) = 0 whenever A, B ∈ F and μ(A) = μ(B) = 0; see [14].
(1) (2) (3) (4)
Definition 3. Let μ : F → E be a non-additive measure. Let {fn } be a sequence of F -measurable real-valued functions on X and f also such a function. (1) {fn } is said to converge μ-a.e. to f if there exists an A ∈ F with μ(A) = 0 such that {fn } converges to f on X A. (2) {fn } is said to converge μ-almost uniformly to f if there exists a decreasing net {Bγ | γ ∈ Γ } ⊂ F such that for any U ∈ B0 , there exists a γ ∈ Γ such that μ(Bγ ) ∈ U and {fn } converges to f uniformly on each subset X Bγ . Definition 4. (See [15].) A double sequence {rm,n } in E is called a topological regulator if it satisfies the following two conditions: (1) rm,n rm,n+1 for any m, n ∈ N . (2) rm,n → 0 for any m ∈ N . Definition 5. (See [15].) E has property (EP) if for any topological regulator {rm,n } in E, there exists a sequence {pk } in E satisfying the following two conditions: (1) pk → 0. (2) For every k ∈ N and m ∈ N , there exists an n0 (m, k) ∈ N such that rm,n pk for any n n0 (m, k). Definition 6. Let μ : F → E be a non-additive measure. (1) A double sequence {Am,n } ⊂ F is called a μ-regulator if it satisfies the following two conditions: (D1) Am,n ⊃ Am,n whenever n n .
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∞ (D2) μ( ∞ m=1 n=1 Am,n ) = 0. (2) μ satisfies the Egoroff condition if for any μ-regulator {Am,n } and U ∈ B0 there exists a θ ∈ Θ such that μ( ∞ m=1 Am,θ(m) ) ∈ U . Remark 1. A non-additive measure μ satisfies the Egoroff condition if (and only if), for any double sequence
{A m,n } ⊂ F satisfying (D2) and the following (D1 ), it holds that for any U ∈ B0 there exists a θ ∈ Θ such that A ) ∈ U . μ( ∞ m=1 m,θ(m) (D1 ) Am,n ⊃ Am ,n whenever m m and n n . 3. The Egoroff condition and regularity of non-additive measures In this section, we give an ordered topological vector space version of the regularity of non-additive measures. This result is connected with [9, Theorem 4.1]. First, we have the following lemma. This is an ordered topological vector space version of [12, Proposition 3]. Lemma 1. Let μ : F → E be a non-additive measure. If μ satisfies the Egoroff condition, then μ is strongly order continuous. Proof. We assume that the Egoroff condition holds. Let {Bn } ⊂ F be a decreasing sequence such that {Bn } converges to a set B ∈ F with μ(B) = 0. Define double sequence {Am,n } ⊂ F as Am,n = Bn for all m, n ∈ N . Since Am ,n ⊂ Am,n for m m and n n , and ∞ ∞ ∞ ∞ μ Am,n = μ Bn = μ(B) = 0, m=1 n=1
m=1 n=1
{Am,n } is a μ-regulator. For any U ∈ B0 take V ∈ B0 such that V ⊂ U and V is full. We apply to {Am,n } the Egoroff condition. Then there exists a θ ∈ Θ such that ∞ ∞ μ(Bn ) μ(Bθ(1) ) = μ Bθ(m) = μ Am,θ(m) ∈ V m=1
m=1
for any n θ (1). Since V is full, we have μ(Bn ) ∈ V ⊂ U. Thus μ is strongly order continuous.
2
Definition 7. Let μ : F → E be a non-additive measure. (1) μ is said to be countably weakly null-additive if μ( ∞ n=1 An ) = 0 whenever {An } ⊂ F and μ(An ) = 0 for any n ∈ N ; see [9]. (2) μ is said to be null-continuous if μ( ∞ n=1 An ) = 0 for every increasing sequence {An } ⊂ F such that μ(An ) = 0 for any n ∈ N ; see [2]. By the above definition, the following proposition holds. This is an ordered topological vector space version of [9, Proposition 2.1]. Proposition 2. (See [9].) Let μ : F → E be a non-additive measure. μ is countably weakly null-additive if and only if μ is both weakly null-additive and null-continuous. The following lemma is an ordered topological vector space version of [2, Lemma 3]. The method of proof belongs to that of [2].
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Lemma 3. (See [2].) Let μ : F → E be a non-additive measure. If μ is weakly null-additive and strongly order continuous, then μ is null-continuous. Proof. Since E is Hausdorff and satisfies the first axiom of countability, there exists a sequence {Um } ⊂ B0 such that ∞ m=1 Um = {0} and Um is full. We may assume that {Um } is decreasing without any loss of generality. We also assume that μ is weakly null-additive and strongly order continuous. Let A ∈ F and a sequence {An } ⊂ F with An A and μ(An ) = 0 for any n ∈ N . We define a subsequence {Ank } of {An } as follows. Let n1 = 1. For every k ∈ N , since μ(Ank ) = 0 and Ank ∪ A An Ank as n → ∞, by the strongly order continuity of μ, we can choose nm ∈ N with nm+1 > nm and we have μ(Anm ∪ A Anm+1 ) ∈ Um . We define B=
∞
(An2k An2k−1 ),
k=1
C = A B = An1 ∪
∞
(An2k+1 An2k ).
k=1
For every k ∈ N , there exists n2k+1 ∈ N with n2k+1 n2k such that μ An2k ∪ (A An2k+1 ) ∈ U2k , and also since B ⊂ An2k ∪ (A An2k+1 ) and U2k is full, it follows that μ(B) ∈ U2k .
Since {Um } is decreasing, we have ∞ k=1 U2k = {0}. Moreover, since U2k is full, we have μ(B) = 0. Similarly, for every k ∈ N , there exists n2k ∈ N with n2k n2k−1 such that μ An2k−1 ∪ (A An2k ) ∈ U2k−1 , and also since C ⊂ An2k−1 ∪ (A An2k ) and U2k−1 is full, it follows that μ(C) ∈ U2k−1 .
Since {Um } is decreasing, we have ∞ k=1 U2k−1 = {0}. Moreover, since U2k−1 is full, we have μ(C) = 0. By the weak null-additivity of μ, we have μ(A) = μ(B ∪ C) = 0. 2 From the above lemma, we also have the following proposition. Proposition 4. Let μ : F → E be a non-additive measure. If μ is weakly null-additive and strongly order continuous, then it is countably weakly null-additive. By Lemma 1, we have the following proposition. This is an ordered topological vector space version of [9, Proposition 3.4]. Proposition 5. Let μ : F → E be a non-additive measure. If μ is weakly null-additive and satisfies the Egoroff condition, then it is null-continuous and countably weakly null-additive. By the Egoroff condition, we have the following proposition. This is also an ordered topological vector space version of [9, Proposition 3.6]. Proposition 6. Let μ : F → E be a non-additive measure. Then the following two conditions are equivalent: (i) μ satisfies the Egoroff condition. (ii) For any U ∈ B0 and double sequence {Am,n } ⊂ Fsatisfying that Am,n Dm as n → ∞ and μ( ∞ m=0 Dm ) = 0 for each m ∈ N , there exists a θ ∈ Θ such that μ( ∞ A ) ∈ U . m,θ(m) m=1 By Propositions 5 and 6, we have the following lemma. This is an ordered topological vector space version of [9, Proposition 3.7].
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Lemma 7. Let μ : F → E be a non-additive measure. Then the following two conditions are equivalent: (i) μ is weakly null-additive and satisfies the Egoroff condition. (ii) For any U ∈ B0 and double sequence {Am,n } ⊂ F satisfying that Am,n Dm as n → ∞ and μ(Dm ) = 0 for each m ∈ N , there exists a θ ∈ Θ such that μ( ∞ m=1 Am,θ(m) ) ∈ U . Let E have property (EP). By Lemma 7, if μ is strongly order continuous and has property (S), then μ satisfies the Egoroff condition. Thus we have the following corollary; see also [16]. Corollary 8. Let μ : F → E be a non-additive measure which is strongly order continuous and has property (S). We assume that E has property (EP). Then the following two conditions are equivalent: (i) μ is weakly null-additive. ⊂ F satisfying that Am,n Dm as n → ∞ and μ(Dm ) = 0 for (ii) For any U ∈ B0 and double sequence {Am,n } each m ∈ N , there exists a θ ∈ Θ such that μ( ∞ m=1 Am,θ(m) ) ∈ U . Definition 8. (See [4].) Let μ : F → E be a non-additive measure. μ is said to have the pseudometric generating property if for any U ∈ B0 , there exists a V ∈ B0 such that for any A, B ∈ F , if μ(A) ∈ V and μ(B) ∈ V , then μ(A ∪ B) ∈ U . We can also prove the following lemma which is an ordered topological vector space version of [9, Proposition 5.1]. Lemma 9. Let μ : F → E be a non-additive measure which is strongly order continuous and has property (S). If μ is weakly null-additive, then it has the pseudometric generating property. Proof. We assume that there exist U ∈ B0 and sequences {An } ⊂ F , {Bn } ⊂ F such that μ(An ) → 0, μ(Bn ) → 0 and / U for any n n0 . Since μ(An ) → 0 and μ(Bn ) → 0, by the property (S), that there exists n0 ∈ N with μ(An ∪ Bn ) ∈ there exist subsequences {Ank } and {Bmk } such that ∞ ∞ ∞ ∞ μ Ank = 0 and μ Bmk = 0. i=1 k=i
i=1 k=i
Since μ is weakly null-additive, we have ∞ ∞ ∞ ∞ ∞ ∞ μ (Ank ∪ Bmk ) = μ Ank ∪ Bmk = 0. i=1 k=i
i=1 k=i
i=1 k=i
For U ∈ B0 there exists a V ∈ B0 such that V ⊂ U and V is full. Since ∞
(Ank ∪ Bmk )
k=i
∞ ∞
(Ank ∪ Bmk ) as i → ∞,
i=1 k=i
and μ is strongly order continuous, there exists j0 ∈ N such that ∞ μ (Ank ∪ Bmk ) ∈ V . k=j0
Since
μ(Anj0 ∪ Bmj0 ) μ
∞
(Ank ∪ Bmk )
k=j0
and V is full, we have μ(Anj ∪ Bmj ) ∈ V ⊂ U
for any j j0 .
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It contradicts the fact that there exists U ∈ B0 such that there exists j0 ∈ N with μ(Anj ∪ Bmj ) ∈ / U for any j j0 .
2
Lemma 10. Let μ : F → E be a non-additive measure. If μ satisfies the pseudometric generating property, then it is weakly null-additive. Proof. Since E is Hausdorff and satisfies the first axiom of countability, there exists {Un } ⊂ B0 such that ∞ n=1 Un = {0} and Un is full. For Un there exists Vn ⊂ B0 such that Vn ⊂ Un . Let A, B ∈ F with μ(A) = 0 and μ(B) = 0. Then we ∞have μ(A) ∈ Vn and μ(B) ∈ Vn . Since μ has the pseudometric generating property, we have μ(A ∪ B) ∈ Un . Since n=1 Un = {0}, we have μ(A ∪ B) = 0. 2 Let X be a Hausdorff space. Denote by B(X) the σ -field of all Borel subsets of X, that is, the σ -field generated by the open subsets of X. A non-additive measure defined on B(X) is called a non-additive Borel measure on X. Definition 9. (See [17].) Let μ be a non-additive Borel measure on X. μ is called regular if for any U ∈ B0 and A ∈ B(X), there exist a closed set FU and an open set GU such that FU ⊂ A ⊂ GU and μ(GU FU ) ∈ U . Now we show our main result. The following theorem is an ordered topological vector space version of [9, Theorem 4.1]. The method of proof belongs to that of [9]. Theorem 11. Let X be a metric space and B(X) a σ -field of all Borel subsets of X. Let μ : B(X) → E be a nonadditive Borel measure on X. If μ has the pseudometric generating property and satisfies the Egoroff condition, then μ is regular. Proof. Let μ : B(X) → E be a non-additive Borel measure. Denote by E the family of Borel subsets A of X with the property that for any U ∈ B0 , there exist a closed set FU and an open set GU such that F U ⊂ A ⊂ GU
and μ(GU FU ) ∈ U.
We first show that E is a σ -field. It is obvious that E is closed for complementation and contains ∅ and X. We show that E is closed for countable unions. Let {Am } be a sequence of E and put A = ∞ m=1 Am on X. Since E is Hausdorff and satisfies the first axiom of countability, there exists {Vn } ⊂ B0 such that ∞ n=1 Vn = {0} and Vn is full. Then for each m ∈ N , there exist double sequences {Fm,n } of closed sets and {Gm,n } of open sets such that Fm,n ⊂ Am ⊂ Gm,n
and μ(Gm,n Fm,n ) ∈ Vn
for all n.
We may assume that, for each m ∈ N , {Fm,n } is increasing and {Gm,n } is decreasing without loss of generality. For each m ∈ N , we put Dm = ∞ n=1 (Gm,n Fm,n ). Since (Gm,n Fm,n ) Dm
as n → ∞
and Vn is full, we have μ(Dm ) ∈ Vn . Moreover, since ∞ n=1 Vn = {0}, we have μ(Dm ) = 0. For any U ∈ B0 , we take V ∈ B0 such that V ⊂ U and V is full, and also take W ∈ B0 such that W ⊂ V . By Lemma 10, μ is weakly null-additive. Then by Lemma 7, there exists a θ ∈ Θ such that ∞ μ (Gm,θ(m) Fm,θ(m) ) ∈ W. m=1
On the other hand by Lemma 1, μ is strongly order continuous. Since ∞ n Fm,θ(m) Fm,θ(m) ∅ m=1
m=1
as n → ∞, there exists N0 such that ∞ N0 μ Fm,θ(m) Fm,θ(m) ∈ W. m=1
m=1
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Since μ has the pseudometric generating property, we have μ
∞
∞
(Gm,θ(m) Fm,θ(m) ) ∪
m=1
Fm,θ(m)
m=1
N0
∈ V.
Fm,θ(m)
m=1
Also since ∞
Gm,θ(m)
m=1
N0
Fm,θ(m) ⊂
m=1
∞
(Gm,θ(m) Fm,θ(m) ) ∪
m=1
∞ m=1
Fm,θ(m)
N0
Fm,θ(m)
m=1
and V is full, we have ∞ N0 μ Gm,θ(m) Fm,θ(m) ∈ V ⊂ U. m=1
Denote FU =
m=1
N0
m=1 Fm,θ(m)
F U ⊂ A ⊂ GU
and GU =
∞
m=1 Gm,θ(m) ,
then FU is closed, GU is open and we have
and μ(GU FU ) ∈ U.
Therefore A ∈ E . Thus E is a σ -field. Next we verify that E contains all closed subsets of X. Let F be closed in X. Since X is a metric space, one can find a sequence {Gn } of open subsets of X such that Gn F ∅, and μ is strongly order continuous, hence we have μ(Gn F ) → 0. Thus, we have F ∈ E . Consequently, E is a σ -field which contains all closed subsets of X, so that it also contains all Borel subsets of X. Therefore μ is regular. 2 Example 1. Let X = [0, 1] be a metric space with the metric d(x, y) = |x − y|, B(X) the σ -field of all Borel subsets of X and m the Lebesgue measure on B(X). Define
a · m(A) if m(A) < 1, μ(A) = 1 if m(A) = 1, where 0 < a < 1. Then μ is a non-additive measure. Since m is the Lebesgue measure, it is easy to see that μ has the pseudometric generating property and satisfies the Egoroff condition. 4. Egoroff’s theorem In this section, we give an ordered topological vector space version of Egoroff’s theorem on a metric space. This result is connected with [9, Theorem 5.2]. The following result is an ordered topological vector space version of [9, Theorem 5.1]; see also [15]. Theorem 12. Let μ : F → E be a non-additive measure. Then the following statements are equivalent: (i) μ satisfies the Egoroff condition. (ii) The Egoroff theorem holds for μ, that is, let {fn } be a sequence of F -measurable real-valued functions on X and f also such a function. If {fn } converges μ-a.e. to f , then {fn } converges μ-almost uniformly to f . By Theorem 12, we have the following. This is an ordered topological vector space version of [9, Corollay 5.1]. Theorem 13. Let μ : B(X) → E be a non-additive Borel measure that satisfies the Egoroff condition. Let {fn } be a sequence of Borel measurable real-valued functions on X and f also such a function. If {fn } converges μ-a.e. to f , then there exists an increasing sequence {Am } ⊂ B(X) such that μ(X ∞ m=1 Am ) = 0 and {fn } converges to f uniformly on Am for each m ∈ N .
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Proof. Since μ satisfies the Egoroff condition and {fn } converges μ-a.e. to f , by Theorem 12, for any U ∈ B0 , there exists a decreasing net {Bγ | γ ∈ Γ } such that μ(Bγ ) ∈ U and {fn } converges to f uniformly on each ∞set X Bγ . Since E is Hausdorff and satisfies the first axiom of countability, there exists {V } ⊂ B such that 0 m=1 Vm = {0} m and Vm is full. Then there exists a {γm } such that μ(Bγm ) ∈ Vm . Put Am = X m B for each m ∈ N . The proof i=1 γi is complete. 2 By Lemma 7, Theorems 11 and 13, we also have the following. This is an ordered topological vector space version of [9, Theorem 5.2]. The method of proof belongs to that of [9]. Theorem 14. Let X be a metric space and μ : B(X) → E a non-additive Borel measure that has the pseudometric generating property and satisfies the Egoroff condition. Let {fn } be a sequence of Borel measurable real-valued functions on X and f also such a function. If {fn } converges μ-a.e. to f , then for any U ∈ B0 , there exists a closed set FU such that μ(X FU ) ∈ U and {fn } converges to f uniformly on each FU . Proof. Since {fn } converges μ-a.e. to f , by Theorem 13, there exists an increasing sequence {Am } ⊂ B(X) such that {fn } converges to f uniformly on Am for each m ∈ N and μ(X ∞ m=1 Am ) = 0. Since E is Hausdorff and satisfies the first axiom of countability, there exists {Vn } ⊂ B0 such that ∞ n=1 Vn = {0} and Vn is full. By Theorem 11, since μ is regular, for each m ∈ N , there exists an increasing sequence {Fm,n } of closed sets such that Fm,n ⊂ Am and μ(Am Fm,n ) ∈ Vn for each n ∈ N . Without loss of generality, we can assume that for each m ∈ N , {Am Fm,n } is decreasing as n → ∞. Then we have (Am Fm,n )
∞
(Am Fm,n )
as n → ∞.
n=1
∞ Put Xm,n = (X ∞ m=1 Am ) ∪ (Am Fm,n ) and Dm = n=1 Xm,n . Then for each m ∈ N , we have Xm,n Dm as n → ∞. Since Vn is full and ∞ μ (Am Fm,n ) μ(Am Fm,n ), n=1
we have μ Since
∞
(Am Fm,n ) ∈ Vn .
n=1 ∞ n=1 Vn ∞
= {0}, we have
(Am Fm,n ) = 0.
μ
n=1
By Lemma 10, since μ is weakly null-additive, we have μ(Dm ) = 0 for each m ∈ N . For any U ∈ B0 , we take V ∈ B0 such that V ⊂ U and V is full, and also take W ∈ B 0∞such that W ⊂V∞and W is full. By Lemma 7, there exists a θ ∈ Θ such that μ( ∞ X ) ∈ W . Since X m,θ(m) m=1 m=1 Fm,θ(m) ⊂ m=1 Xm,θ(m) and W is full, we have ∞ Fm,θ(m) ∈ W. μ X m=1
On the other hand by Lemma 1, μ is strongly order continuous. Since ∞ n Fm,θ(m) Fm,θ(m) ∅ m=1
m=1
as n → ∞, there exists an N0 ∈ N such that ∞ N0 Fm,θ(m) Fm,θ(m) ∈ W. μ m=1
m=1
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Since μ has the pseudometric generating property, we have ∞ N0 ∞ μ X Fm,θ(m) ∪ Fm,θ(m) Fm,θ(m) ∈ V. m=1
Also since X
N0
m=1
Fm,θ(m) ⊂ X
m=1
∞
m=1
Fm,θ(m) ∪
m=1
∞ m=1
Fm,θ(m)
N0
Fm,θ(m)
m=1
and V is full, we have N0 Fm,θ(m) ∈ V ⊂ U. μ X m=1
N0
Denote FU = m=1 Fm,θ(m) , then FU is a closed set, μ(X FU ) ∈ U and FU ⊂ converges to f uniformly on FU . 2
N
m=1 Am .
It is easy to see that {fn }
5. Lusin’s theorem In this section, by using the results obtained in Sections 3 and 4, we prove an ordered topological vector space version of Lusin’s theorem. This result is connected with [9, Theorem 5.3]. The method of proof belongs to that of [9]. Theorem 15. Let X be a metric space and μ : B(X) → E a non-additive Borel measure on X that has the pseudometric generating property and satisfies the Egoroff condition. If f is a Borel measurable real-valued function on X, then for any U ∈ B0 , there exists a closed set FU such that μ(X FU ) ∈ U and f is continuous on each FU . Proof. Since μ has the pseudometric generating property, by Lemma 10, μ is weakly null-additive. By using Lemma 7 and Theorem 14, the proof is straightforward to that of [16, Theorem 6]. 2 Remark 2. For real-valued fuzzy measures, see [11, Theorem 4], and for Riesz space-valued fuzzy measures, see [7, Theorem 3]. By Lemma 9, Theorem 15 and [15, Theorem 3], we have the following corollary which is an ordered topological vector space version of [9, Corollary 6.3]. Corollary 16. Let X be a metric space and μ : B(X) → E a non-additive Borel measure on X which is strongly order continuous, property (S), and has the pseudometric generating property. We assume that E has property (EP). If f is a Borel measurable real-valued function on X, then for any U ∈ B0 , there exists a closed set FU such that μ(X FU ) ∈ U and f is continuous on each FU . By Theorem 15 and [15, Theorem 4], we have the following corollary which is an ordered topological vector space version of [10, Theorem 4]. Corollary 17. Let X be a metric space and μ : B(X) → E a fuzzy Borel measure on X that has the pseudometric generating property. We assume that E is locally convex. If f is a Borel measurable real-valued function on X, then for any U ∈ B0 , there exists a closed set FU such that μ(X FU ) ∈ U and f is continuous on each FU . 6. Applications In this section, we apply our results in Sections 3, 4 and 5 to the following ordered topological vector spaces.
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(i) Let T be a Hausdorff space and C(T ) the space of all real continuous functions defined on T endowed with a pointwise order. Then C(T ) is an ordered vector space. The topology of compact convergence on C(T ) is a locally full topology; see [3, p. 159]. Moreover, let T be a locally compact space which is the union of a countable family of compact sets. Then its topology is defined by the sequence of semi-norms. Thus it is metrizable. For any metrizable topological vector space, there exists a countable neighborhoods of the origin in it; see [3, p. 40]. Thus the first axiom of countability holds. Clearly C(T ) is Hausdorff. (ii) Let C ∞ (R n ) be the space of all real functions having continuous derivatives of any order. C ∞ (R n ) is an ordered locally convex space but not a Riesz space; see [3, p. 159]. Similar to (i), it has the first axiom of countability. Clearly it is also Hausdorff. (iii) Let Lp ([0, 1]) be the space of real-valued Lebesgue measurable functions f defined on [0, 1] such that 1 p 0 |f (x)| dμ < ∞ (0 < p < ∞) endowed with the pointwise order. Then Lp ([0, 1]) is a Riesz space. Moreover, Lp ([0, 1]) is a locally solid space; see [1, example 8.6]. If the topology on a Riesz space is locally solid, then it is locally full, see [3, Chapter 7, Proposition 1]. Therefore Lp ([0, 1]) is an ordered topological vector space. However, its topology is not locally convex if 0 < p < 1. Since its topology is defined by a unique quasi-norm, it is metrizable. Thus it satisfies the first axiom of countability. Acknowledgements The authors would like to express their hearty thanks to anonymous referee for careful readings and valuable suggestions. We would like to express our hearty thanks to Professor Shizu Nakanishi. We also would like to thank Professor Toshiharu Kawasaki and Professor Ichiro Suzuki for many valuable suggestions. References [1] C. Aliprantis, O. Burkinshaw, Locally Solid Spaces, Academic Press, New York, 1978. [2] S. Asahina, K. Uchino, T. Murofushi, Relationship among continuity and null-additive conditions in non-additive measure theory, Fuzzy Sets Syst. 157 (2006) 691–698. [3] R. Cristescu, Topological Vector Spaces, Noordhoff International Publishing, Leyden, 1977. [4] I. Dobrakov, J. Farkova, On submeasures II, Math. Slovaca 30 (1980) 65–81. [5] Q. Jiang, H. Suzuki, Fuzzy measures on metric spaces, Fuzzy Sets Syst. 83 (1996) 99–106. [6] Q. Jiang, S. Wang, D. Ziou, A further investigation for fuzzy measures on metric spaces, Fuzzy Sets Syst. 105 (1999) 293–297. [7] J. Kawabe, Regularity and Lusin’s theorem for Riesz space-valued fuzzy measures, Fuzzy Sets Syst. 158 (2007) 895–903. [8] J. Li, Order continuous of monotone set function and convergence of measurable functions sequence, Appl. Math. Comput. 135 (2003) 211–218. [9] J. Li, R. Mesiar, Lusin’s theorem on monotone measure spaces, Fuzzy Sets Syst. 175 (2011) 75–86. [10] J. Li, M. Yasuda, Egoroff’s theorems on monotone non-additive measure space, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 12 (2004) 61–68. [11] J. Li, M. Yasuda, Lusin’s theorems on fuzzy measure spaces, Fuzzy Sets Syst. 146 (2004) 121–133. [12] T. Murofushi, K. Uchino, S. Asahina, Conditions for Egoroff’s theorem in non-additive measure theory, Fuzzy Sets Syst. 146 (2004) 135–146. [13] Q. Sun, Property (S) of fuzzy measure and Riesz’s theorem, Fuzzy Sets Syst. 62 (1994) 117–119. [14] Z. Wang, G.J. Klir, Fuzzy Measure Theory, Plenum Press, New York, 1992. [15] T. Watanabe, On sufficient conditions for the Egoroff theorem of an ordered topological vector space-valued non-additive measure, Fuzzy Sets Syst. 162 (2011) 79–83. [16] T. Watanabe, T. Kawasaki, T. Tanaka, On a sufficient condition of Lusin’s theorem for non-additive measures that take values in an ordered topological vector space, Fuzzy Sets Syst. 194 (2012) 66–75. [17] C. Wu, M. Ha, On the regularity of the fuzzy measure on metric fuzzy measure spaces, Fuzzy Sets Syst. 66 (1994) 373–379.
ScienceDirect Fuzzy Sets and Systems 244 (2014) 41–50 www.elsevier.com/locate/fss
On Lusin’s theorem for non-additive measures that take values in an ordered topological vector space Toshikazu Watanabe ∗ , Tamaki Tanaka Graduate School of Science and Technology, Niigata University, 8050, Ikarashi 2-no-cho, Nishi-ku, Niigata, 950-2181, Japan Received 1 December 2011; received in revised form 16 July 2013; accepted 26 July 2013 Available online 6 August 2013
Abstract Lusin’s theorem was established for real-valued monotone measures under an equivalent condition to Egoroff’s theorem recently. In this paper, we show that the same result remains valid for non-additive measures that take values in an ordered topological vector space. We apply our result to several ordered topological vector spaces. © 2013 Elsevier B.V. All rights reserved. Keywords: Non-additive measure; Egoroff’s theorem; Lusin’s theorem; Ordered vector space; Locally full topology; Pseudometric generating property
1. Introduction The regularity of measure on a topological space is an important property in the measure theory. In fact, Lusin’s theorem can be discussed using such arguments in a metric space. For real-valued fuzzy measure, in [17], Wu and Ha generalized Lusin’s theorem from a classical measure space to a finite autocontinuous fuzzy measure space. Jiang et al. [5,6] extended the result of Wu and Ha. Li and Yasuda showed Lusin’s theorem for real-valued fuzzy measures on a metric space in [11]. Kawabe proved Lusin’s theorem for Riesz space-valued fuzzy Borel measures; see [7]. In [16], the authors also proved Lusin’s theorem for non-additive Borel measures which take values in an ordered topological vector space under the following assumptions; the measure is continuous from above with a property suggested by Sun [13] and weakly null-additive, and the image space has an ordered topological vector space version of the Egoroff property. Recently in [9], Li and Mesiar proved Lusin’s theorem for real-valued monotone measures on a metric space under an equivalent condition to Egoroff’s theorem by using the pseudometric generating property of set function. In this paper, motivated by [9], we prove Lusin’s theorem for non-additive Borel measures which take values in an ordered topological vector space under the measure has the pseudometric generating property and satisfies the Egoroff condition. We apply our result to several ordered topological vector spaces. * Corresponding author.
E-mail addresses: [email protected] (T. Watanabe), [email protected] (T. Tanaka). 0165-0114/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2013.07.021
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2. Preliminaries Let R be the set of real numbers and N the set of natural numbers. Denote by Θ the set of all mappings from N into N . Let X be a non-empty set and F a σ -field of X. A topology on a vector space E is called a vector topology if the mappings (x, y) → x + y and (α, x) → αx are continuous for any x, y ∈ E and α ∈ R. A subset F of an ordered vector space E is called full if x1 , x2 ∈ F and x1 x2 implies [x1 , x2 ] = {x ∈ E | x1 x x2 } ⊂ F . We consider a vector topology on E and let B0 be a system of neighborhoods of 0 ∈ E. The vector topology on E is called locally full, if there exists a basis of B0 consisting of full sets. An ordered vector space endowed with this topology is called an ordered topological vector space; see [3]. We also assume that E is a Hausdorff space with the first axiom of countability. Let {un } be a sequence in E and u ∈ E. We say that un converges to u and write un → u if for any U ∈ B0 , there exists an n0 ∈ N such that un − u ∈ U for any n n0 . Definition 1. A set function μ : F → E is called a non-additive measure if it satisfies the following two conditions: (1) μ(∅) = 0. (2) If A, B ∈ F and A ⊂ B, then μ(A) μ(B). Definition 2. Let μ : F → E be a non-additive measure. μ is said to be continuous from above if μ(An ) → μ(A) whenever {An } ⊂ F and A ∈ F satisfy An A. μ is said to be continuous from below if μ(An ) → μ(A) whenever {An } ⊂ F and A ∈ F satisfy An A. μ is called a fuzzy measure if it is continuous from above and below. μ is said to be strongly order continuous if it is continuous from above at measurable sets of measure 0, that is, for any {An } ⊂ F and A ∈ F with An A and μ(A) = 0, it holds that μ(An ) → 0; see [8]. (5) μ has ∞ (S) if for any sequence {An } ⊂ F with μ(An ) → 0, there exists a subsequence {Ank } such that property μ( ∞ i=1 k=i Ank ) = 0; see [13]. (6) μ is said to be weakly null-additive if μ(A ∪ B) = 0 whenever A, B ∈ F and μ(A) = μ(B) = 0; see [14].
(1) (2) (3) (4)
Definition 3. Let μ : F → E be a non-additive measure. Let {fn } be a sequence of F -measurable real-valued functions on X and f also such a function. (1) {fn } is said to converge μ-a.e. to f if there exists an A ∈ F with μ(A) = 0 such that {fn } converges to f on X A. (2) {fn } is said to converge μ-almost uniformly to f if there exists a decreasing net {Bγ | γ ∈ Γ } ⊂ F such that for any U ∈ B0 , there exists a γ ∈ Γ such that μ(Bγ ) ∈ U and {fn } converges to f uniformly on each subset X Bγ . Definition 4. (See [15].) A double sequence {rm,n } in E is called a topological regulator if it satisfies the following two conditions: (1) rm,n rm,n+1 for any m, n ∈ N . (2) rm,n → 0 for any m ∈ N . Definition 5. (See [15].) E has property (EP) if for any topological regulator {rm,n } in E, there exists a sequence {pk } in E satisfying the following two conditions: (1) pk → 0. (2) For every k ∈ N and m ∈ N , there exists an n0 (m, k) ∈ N such that rm,n pk for any n n0 (m, k). Definition 6. Let μ : F → E be a non-additive measure. (1) A double sequence {Am,n } ⊂ F is called a μ-regulator if it satisfies the following two conditions: (D1) Am,n ⊃ Am,n whenever n n .
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∞ (D2) μ( ∞ m=1 n=1 Am,n ) = 0. (2) μ satisfies the Egoroff condition if for any μ-regulator {Am,n } and U ∈ B0 there exists a θ ∈ Θ such that μ( ∞ m=1 Am,θ(m) ) ∈ U . Remark 1. A non-additive measure μ satisfies the Egoroff condition if (and only if), for any double sequence
{A m,n } ⊂ F satisfying (D2) and the following (D1 ), it holds that for any U ∈ B0 there exists a θ ∈ Θ such that A ) ∈ U . μ( ∞ m=1 m,θ(m) (D1 ) Am,n ⊃ Am ,n whenever m m and n n . 3. The Egoroff condition and regularity of non-additive measures In this section, we give an ordered topological vector space version of the regularity of non-additive measures. This result is connected with [9, Theorem 4.1]. First, we have the following lemma. This is an ordered topological vector space version of [12, Proposition 3]. Lemma 1. Let μ : F → E be a non-additive measure. If μ satisfies the Egoroff condition, then μ is strongly order continuous. Proof. We assume that the Egoroff condition holds. Let {Bn } ⊂ F be a decreasing sequence such that {Bn } converges to a set B ∈ F with μ(B) = 0. Define double sequence {Am,n } ⊂ F as Am,n = Bn for all m, n ∈ N . Since Am ,n ⊂ Am,n for m m and n n , and ∞ ∞ ∞ ∞ μ Am,n = μ Bn = μ(B) = 0, m=1 n=1
m=1 n=1
{Am,n } is a μ-regulator. For any U ∈ B0 take V ∈ B0 such that V ⊂ U and V is full. We apply to {Am,n } the Egoroff condition. Then there exists a θ ∈ Θ such that ∞ ∞ μ(Bn ) μ(Bθ(1) ) = μ Bθ(m) = μ Am,θ(m) ∈ V m=1
m=1
for any n θ (1). Since V is full, we have μ(Bn ) ∈ V ⊂ U. Thus μ is strongly order continuous.
2
Definition 7. Let μ : F → E be a non-additive measure. (1) μ is said to be countably weakly null-additive if μ( ∞ n=1 An ) = 0 whenever {An } ⊂ F and μ(An ) = 0 for any n ∈ N ; see [9]. (2) μ is said to be null-continuous if μ( ∞ n=1 An ) = 0 for every increasing sequence {An } ⊂ F such that μ(An ) = 0 for any n ∈ N ; see [2]. By the above definition, the following proposition holds. This is an ordered topological vector space version of [9, Proposition 2.1]. Proposition 2. (See [9].) Let μ : F → E be a non-additive measure. μ is countably weakly null-additive if and only if μ is both weakly null-additive and null-continuous. The following lemma is an ordered topological vector space version of [2, Lemma 3]. The method of proof belongs to that of [2].
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Lemma 3. (See [2].) Let μ : F → E be a non-additive measure. If μ is weakly null-additive and strongly order continuous, then μ is null-continuous. Proof. Since E is Hausdorff and satisfies the first axiom of countability, there exists a sequence {Um } ⊂ B0 such that ∞ m=1 Um = {0} and Um is full. We may assume that {Um } is decreasing without any loss of generality. We also assume that μ is weakly null-additive and strongly order continuous. Let A ∈ F and a sequence {An } ⊂ F with An A and μ(An ) = 0 for any n ∈ N . We define a subsequence {Ank } of {An } as follows. Let n1 = 1. For every k ∈ N , since μ(Ank ) = 0 and Ank ∪ A An Ank as n → ∞, by the strongly order continuity of μ, we can choose nm ∈ N with nm+1 > nm and we have μ(Anm ∪ A Anm+1 ) ∈ Um . We define B=
∞
(An2k An2k−1 ),
k=1
C = A B = An1 ∪
∞
(An2k+1 An2k ).
k=1
For every k ∈ N , there exists n2k+1 ∈ N with n2k+1 n2k such that μ An2k ∪ (A An2k+1 ) ∈ U2k , and also since B ⊂ An2k ∪ (A An2k+1 ) and U2k is full, it follows that μ(B) ∈ U2k .
Since {Um } is decreasing, we have ∞ k=1 U2k = {0}. Moreover, since U2k is full, we have μ(B) = 0. Similarly, for every k ∈ N , there exists n2k ∈ N with n2k n2k−1 such that μ An2k−1 ∪ (A An2k ) ∈ U2k−1 , and also since C ⊂ An2k−1 ∪ (A An2k ) and U2k−1 is full, it follows that μ(C) ∈ U2k−1 .
Since {Um } is decreasing, we have ∞ k=1 U2k−1 = {0}. Moreover, since U2k−1 is full, we have μ(C) = 0. By the weak null-additivity of μ, we have μ(A) = μ(B ∪ C) = 0. 2 From the above lemma, we also have the following proposition. Proposition 4. Let μ : F → E be a non-additive measure. If μ is weakly null-additive and strongly order continuous, then it is countably weakly null-additive. By Lemma 1, we have the following proposition. This is an ordered topological vector space version of [9, Proposition 3.4]. Proposition 5. Let μ : F → E be a non-additive measure. If μ is weakly null-additive and satisfies the Egoroff condition, then it is null-continuous and countably weakly null-additive. By the Egoroff condition, we have the following proposition. This is also an ordered topological vector space version of [9, Proposition 3.6]. Proposition 6. Let μ : F → E be a non-additive measure. Then the following two conditions are equivalent: (i) μ satisfies the Egoroff condition. (ii) For any U ∈ B0 and double sequence {Am,n } ⊂ Fsatisfying that Am,n Dm as n → ∞ and μ( ∞ m=0 Dm ) = 0 for each m ∈ N , there exists a θ ∈ Θ such that μ( ∞ A ) ∈ U . m,θ(m) m=1 By Propositions 5 and 6, we have the following lemma. This is an ordered topological vector space version of [9, Proposition 3.7].
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Lemma 7. Let μ : F → E be a non-additive measure. Then the following two conditions are equivalent: (i) μ is weakly null-additive and satisfies the Egoroff condition. (ii) For any U ∈ B0 and double sequence {Am,n } ⊂ F satisfying that Am,n Dm as n → ∞ and μ(Dm ) = 0 for each m ∈ N , there exists a θ ∈ Θ such that μ( ∞ m=1 Am,θ(m) ) ∈ U . Let E have property (EP). By Lemma 7, if μ is strongly order continuous and has property (S), then μ satisfies the Egoroff condition. Thus we have the following corollary; see also [16]. Corollary 8. Let μ : F → E be a non-additive measure which is strongly order continuous and has property (S). We assume that E has property (EP). Then the following two conditions are equivalent: (i) μ is weakly null-additive. ⊂ F satisfying that Am,n Dm as n → ∞ and μ(Dm ) = 0 for (ii) For any U ∈ B0 and double sequence {Am,n } each m ∈ N , there exists a θ ∈ Θ such that μ( ∞ m=1 Am,θ(m) ) ∈ U . Definition 8. (See [4].) Let μ : F → E be a non-additive measure. μ is said to have the pseudometric generating property if for any U ∈ B0 , there exists a V ∈ B0 such that for any A, B ∈ F , if μ(A) ∈ V and μ(B) ∈ V , then μ(A ∪ B) ∈ U . We can also prove the following lemma which is an ordered topological vector space version of [9, Proposition 5.1]. Lemma 9. Let μ : F → E be a non-additive measure which is strongly order continuous and has property (S). If μ is weakly null-additive, then it has the pseudometric generating property. Proof. We assume that there exist U ∈ B0 and sequences {An } ⊂ F , {Bn } ⊂ F such that μ(An ) → 0, μ(Bn ) → 0 and / U for any n n0 . Since μ(An ) → 0 and μ(Bn ) → 0, by the property (S), that there exists n0 ∈ N with μ(An ∪ Bn ) ∈ there exist subsequences {Ank } and {Bmk } such that ∞ ∞ ∞ ∞ μ Ank = 0 and μ Bmk = 0. i=1 k=i
i=1 k=i
Since μ is weakly null-additive, we have ∞ ∞ ∞ ∞ ∞ ∞ μ (Ank ∪ Bmk ) = μ Ank ∪ Bmk = 0. i=1 k=i
i=1 k=i
i=1 k=i
For U ∈ B0 there exists a V ∈ B0 such that V ⊂ U and V is full. Since ∞
(Ank ∪ Bmk )
k=i
∞ ∞
(Ank ∪ Bmk ) as i → ∞,
i=1 k=i
and μ is strongly order continuous, there exists j0 ∈ N such that ∞ μ (Ank ∪ Bmk ) ∈ V . k=j0
Since
μ(Anj0 ∪ Bmj0 ) μ
∞
(Ank ∪ Bmk )
k=j0
and V is full, we have μ(Anj ∪ Bmj ) ∈ V ⊂ U
for any j j0 .
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It contradicts the fact that there exists U ∈ B0 such that there exists j0 ∈ N with μ(Anj ∪ Bmj ) ∈ / U for any j j0 .
2
Lemma 10. Let μ : F → E be a non-additive measure. If μ satisfies the pseudometric generating property, then it is weakly null-additive. Proof. Since E is Hausdorff and satisfies the first axiom of countability, there exists {Un } ⊂ B0 such that ∞ n=1 Un = {0} and Un is full. For Un there exists Vn ⊂ B0 such that Vn ⊂ Un . Let A, B ∈ F with μ(A) = 0 and μ(B) = 0. Then we ∞have μ(A) ∈ Vn and μ(B) ∈ Vn . Since μ has the pseudometric generating property, we have μ(A ∪ B) ∈ Un . Since n=1 Un = {0}, we have μ(A ∪ B) = 0. 2 Let X be a Hausdorff space. Denote by B(X) the σ -field of all Borel subsets of X, that is, the σ -field generated by the open subsets of X. A non-additive measure defined on B(X) is called a non-additive Borel measure on X. Definition 9. (See [17].) Let μ be a non-additive Borel measure on X. μ is called regular if for any U ∈ B0 and A ∈ B(X), there exist a closed set FU and an open set GU such that FU ⊂ A ⊂ GU and μ(GU FU ) ∈ U . Now we show our main result. The following theorem is an ordered topological vector space version of [9, Theorem 4.1]. The method of proof belongs to that of [9]. Theorem 11. Let X be a metric space and B(X) a σ -field of all Borel subsets of X. Let μ : B(X) → E be a nonadditive Borel measure on X. If μ has the pseudometric generating property and satisfies the Egoroff condition, then μ is regular. Proof. Let μ : B(X) → E be a non-additive Borel measure. Denote by E the family of Borel subsets A of X with the property that for any U ∈ B0 , there exist a closed set FU and an open set GU such that F U ⊂ A ⊂ GU
and μ(GU FU ) ∈ U.
We first show that E is a σ -field. It is obvious that E is closed for complementation and contains ∅ and X. We show that E is closed for countable unions. Let {Am } be a sequence of E and put A = ∞ m=1 Am on X. Since E is Hausdorff and satisfies the first axiom of countability, there exists {Vn } ⊂ B0 such that ∞ n=1 Vn = {0} and Vn is full. Then for each m ∈ N , there exist double sequences {Fm,n } of closed sets and {Gm,n } of open sets such that Fm,n ⊂ Am ⊂ Gm,n
and μ(Gm,n Fm,n ) ∈ Vn
for all n.
We may assume that, for each m ∈ N , {Fm,n } is increasing and {Gm,n } is decreasing without loss of generality. For each m ∈ N , we put Dm = ∞ n=1 (Gm,n Fm,n ). Since (Gm,n Fm,n ) Dm
as n → ∞
and Vn is full, we have μ(Dm ) ∈ Vn . Moreover, since ∞ n=1 Vn = {0}, we have μ(Dm ) = 0. For any U ∈ B0 , we take V ∈ B0 such that V ⊂ U and V is full, and also take W ∈ B0 such that W ⊂ V . By Lemma 10, μ is weakly null-additive. Then by Lemma 7, there exists a θ ∈ Θ such that ∞ μ (Gm,θ(m) Fm,θ(m) ) ∈ W. m=1
On the other hand by Lemma 1, μ is strongly order continuous. Since ∞ n Fm,θ(m) Fm,θ(m) ∅ m=1
m=1
as n → ∞, there exists N0 such that ∞ N0 μ Fm,θ(m) Fm,θ(m) ∈ W. m=1
m=1
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Since μ has the pseudometric generating property, we have μ
∞
∞
(Gm,θ(m) Fm,θ(m) ) ∪
m=1
Fm,θ(m)
m=1
N0
∈ V.
Fm,θ(m)
m=1
Also since ∞
Gm,θ(m)
m=1
N0
Fm,θ(m) ⊂
m=1
∞
(Gm,θ(m) Fm,θ(m) ) ∪
m=1
∞ m=1
Fm,θ(m)
N0
Fm,θ(m)
m=1
and V is full, we have ∞ N0 μ Gm,θ(m) Fm,θ(m) ∈ V ⊂ U. m=1
Denote FU =
m=1
N0
m=1 Fm,θ(m)
F U ⊂ A ⊂ GU
and GU =
∞
m=1 Gm,θ(m) ,
then FU is closed, GU is open and we have
and μ(GU FU ) ∈ U.
Therefore A ∈ E . Thus E is a σ -field. Next we verify that E contains all closed subsets of X. Let F be closed in X. Since X is a metric space, one can find a sequence {Gn } of open subsets of X such that Gn F ∅, and μ is strongly order continuous, hence we have μ(Gn F ) → 0. Thus, we have F ∈ E . Consequently, E is a σ -field which contains all closed subsets of X, so that it also contains all Borel subsets of X. Therefore μ is regular. 2 Example 1. Let X = [0, 1] be a metric space with the metric d(x, y) = |x − y|, B(X) the σ -field of all Borel subsets of X and m the Lebesgue measure on B(X). Define
a · m(A) if m(A) < 1, μ(A) = 1 if m(A) = 1, where 0 < a < 1. Then μ is a non-additive measure. Since m is the Lebesgue measure, it is easy to see that μ has the pseudometric generating property and satisfies the Egoroff condition. 4. Egoroff’s theorem In this section, we give an ordered topological vector space version of Egoroff’s theorem on a metric space. This result is connected with [9, Theorem 5.2]. The following result is an ordered topological vector space version of [9, Theorem 5.1]; see also [15]. Theorem 12. Let μ : F → E be a non-additive measure. Then the following statements are equivalent: (i) μ satisfies the Egoroff condition. (ii) The Egoroff theorem holds for μ, that is, let {fn } be a sequence of F -measurable real-valued functions on X and f also such a function. If {fn } converges μ-a.e. to f , then {fn } converges μ-almost uniformly to f . By Theorem 12, we have the following. This is an ordered topological vector space version of [9, Corollay 5.1]. Theorem 13. Let μ : B(X) → E be a non-additive Borel measure that satisfies the Egoroff condition. Let {fn } be a sequence of Borel measurable real-valued functions on X and f also such a function. If {fn } converges μ-a.e. to f , then there exists an increasing sequence {Am } ⊂ B(X) such that μ(X ∞ m=1 Am ) = 0 and {fn } converges to f uniformly on Am for each m ∈ N .
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Proof. Since μ satisfies the Egoroff condition and {fn } converges μ-a.e. to f , by Theorem 12, for any U ∈ B0 , there exists a decreasing net {Bγ | γ ∈ Γ } such that μ(Bγ ) ∈ U and {fn } converges to f uniformly on each ∞set X Bγ . Since E is Hausdorff and satisfies the first axiom of countability, there exists {V } ⊂ B such that 0 m=1 Vm = {0} m and Vm is full. Then there exists a {γm } such that μ(Bγm ) ∈ Vm . Put Am = X m B for each m ∈ N . The proof i=1 γi is complete. 2 By Lemma 7, Theorems 11 and 13, we also have the following. This is an ordered topological vector space version of [9, Theorem 5.2]. The method of proof belongs to that of [9]. Theorem 14. Let X be a metric space and μ : B(X) → E a non-additive Borel measure that has the pseudometric generating property and satisfies the Egoroff condition. Let {fn } be a sequence of Borel measurable real-valued functions on X and f also such a function. If {fn } converges μ-a.e. to f , then for any U ∈ B0 , there exists a closed set FU such that μ(X FU ) ∈ U and {fn } converges to f uniformly on each FU . Proof. Since {fn } converges μ-a.e. to f , by Theorem 13, there exists an increasing sequence {Am } ⊂ B(X) such that {fn } converges to f uniformly on Am for each m ∈ N and μ(X ∞ m=1 Am ) = 0. Since E is Hausdorff and satisfies the first axiom of countability, there exists {Vn } ⊂ B0 such that ∞ n=1 Vn = {0} and Vn is full. By Theorem 11, since μ is regular, for each m ∈ N , there exists an increasing sequence {Fm,n } of closed sets such that Fm,n ⊂ Am and μ(Am Fm,n ) ∈ Vn for each n ∈ N . Without loss of generality, we can assume that for each m ∈ N , {Am Fm,n } is decreasing as n → ∞. Then we have (Am Fm,n )
∞
(Am Fm,n )
as n → ∞.
n=1
∞ Put Xm,n = (X ∞ m=1 Am ) ∪ (Am Fm,n ) and Dm = n=1 Xm,n . Then for each m ∈ N , we have Xm,n Dm as n → ∞. Since Vn is full and ∞ μ (Am Fm,n ) μ(Am Fm,n ), n=1
we have μ Since
∞
(Am Fm,n ) ∈ Vn .
n=1 ∞ n=1 Vn ∞
= {0}, we have
(Am Fm,n ) = 0.
μ
n=1
By Lemma 10, since μ is weakly null-additive, we have μ(Dm ) = 0 for each m ∈ N . For any U ∈ B0 , we take V ∈ B0 such that V ⊂ U and V is full, and also take W ∈ B 0∞such that W ⊂V∞and W is full. By Lemma 7, there exists a θ ∈ Θ such that μ( ∞ X ) ∈ W . Since X m,θ(m) m=1 m=1 Fm,θ(m) ⊂ m=1 Xm,θ(m) and W is full, we have ∞ Fm,θ(m) ∈ W. μ X m=1
On the other hand by Lemma 1, μ is strongly order continuous. Since ∞ n Fm,θ(m) Fm,θ(m) ∅ m=1
m=1
as n → ∞, there exists an N0 ∈ N such that ∞ N0 Fm,θ(m) Fm,θ(m) ∈ W. μ m=1
m=1
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Since μ has the pseudometric generating property, we have ∞ N0 ∞ μ X Fm,θ(m) ∪ Fm,θ(m) Fm,θ(m) ∈ V. m=1
Also since X
N0
m=1
Fm,θ(m) ⊂ X
m=1
∞
m=1
Fm,θ(m) ∪
m=1
∞ m=1
Fm,θ(m)
N0
Fm,θ(m)
m=1
and V is full, we have N0 Fm,θ(m) ∈ V ⊂ U. μ X m=1
N0
Denote FU = m=1 Fm,θ(m) , then FU is a closed set, μ(X FU ) ∈ U and FU ⊂ converges to f uniformly on FU . 2
N
m=1 Am .
It is easy to see that {fn }
5. Lusin’s theorem In this section, by using the results obtained in Sections 3 and 4, we prove an ordered topological vector space version of Lusin’s theorem. This result is connected with [9, Theorem 5.3]. The method of proof belongs to that of [9]. Theorem 15. Let X be a metric space and μ : B(X) → E a non-additive Borel measure on X that has the pseudometric generating property and satisfies the Egoroff condition. If f is a Borel measurable real-valued function on X, then for any U ∈ B0 , there exists a closed set FU such that μ(X FU ) ∈ U and f is continuous on each FU . Proof. Since μ has the pseudometric generating property, by Lemma 10, μ is weakly null-additive. By using Lemma 7 and Theorem 14, the proof is straightforward to that of [16, Theorem 6]. 2 Remark 2. For real-valued fuzzy measures, see [11, Theorem 4], and for Riesz space-valued fuzzy measures, see [7, Theorem 3]. By Lemma 9, Theorem 15 and [15, Theorem 3], we have the following corollary which is an ordered topological vector space version of [9, Corollary 6.3]. Corollary 16. Let X be a metric space and μ : B(X) → E a non-additive Borel measure on X which is strongly order continuous, property (S), and has the pseudometric generating property. We assume that E has property (EP). If f is a Borel measurable real-valued function on X, then for any U ∈ B0 , there exists a closed set FU such that μ(X FU ) ∈ U and f is continuous on each FU . By Theorem 15 and [15, Theorem 4], we have the following corollary which is an ordered topological vector space version of [10, Theorem 4]. Corollary 17. Let X be a metric space and μ : B(X) → E a fuzzy Borel measure on X that has the pseudometric generating property. We assume that E is locally convex. If f is a Borel measurable real-valued function on X, then for any U ∈ B0 , there exists a closed set FU such that μ(X FU ) ∈ U and f is continuous on each FU . 6. Applications In this section, we apply our results in Sections 3, 4 and 5 to the following ordered topological vector spaces.
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(i) Let T be a Hausdorff space and C(T ) the space of all real continuous functions defined on T endowed with a pointwise order. Then C(T ) is an ordered vector space. The topology of compact convergence on C(T ) is a locally full topology; see [3, p. 159]. Moreover, let T be a locally compact space which is the union of a countable family of compact sets. Then its topology is defined by the sequence of semi-norms. Thus it is metrizable. For any metrizable topological vector space, there exists a countable neighborhoods of the origin in it; see [3, p. 40]. Thus the first axiom of countability holds. Clearly C(T ) is Hausdorff. (ii) Let C ∞ (R n ) be the space of all real functions having continuous derivatives of any order. C ∞ (R n ) is an ordered locally convex space but not a Riesz space; see [3, p. 159]. Similar to (i), it has the first axiom of countability. Clearly it is also Hausdorff. (iii) Let Lp ([0, 1]) be the space of real-valued Lebesgue measurable functions f defined on [0, 1] such that 1 p 0 |f (x)| dμ < ∞ (0 < p < ∞) endowed with the pointwise order. Then Lp ([0, 1]) is a Riesz space. Moreover, Lp ([0, 1]) is a locally solid space; see [1, example 8.6]. If the topology on a Riesz space is locally solid, then it is locally full, see [3, Chapter 7, Proposition 1]. Therefore Lp ([0, 1]) is an ordered topological vector space. However, its topology is not locally convex if 0 < p < 1. Since its topology is defined by a unique quasi-norm, it is metrizable. Thus it satisfies the first axiom of countability. Acknowledgements The authors would like to express their hearty thanks to anonymous referee for careful readings and valuable suggestions. We would like to express our hearty thanks to Professor Shizu Nakanishi. We also would like to thank Professor Toshiharu Kawasaki and Professor Ichiro Suzuki for many valuable suggestions. References [1] C. Aliprantis, O. Burkinshaw, Locally Solid Spaces, Academic Press, New York, 1978. [2] S. Asahina, K. Uchino, T. Murofushi, Relationship among continuity and null-additive conditions in non-additive measure theory, Fuzzy Sets Syst. 157 (2006) 691–698. [3] R. Cristescu, Topological Vector Spaces, Noordhoff International Publishing, Leyden, 1977. [4] I. Dobrakov, J. Farkova, On submeasures II, Math. Slovaca 30 (1980) 65–81. [5] Q. Jiang, H. Suzuki, Fuzzy measures on metric spaces, Fuzzy Sets Syst. 83 (1996) 99–106. [6] Q. Jiang, S. Wang, D. Ziou, A further investigation for fuzzy measures on metric spaces, Fuzzy Sets Syst. 105 (1999) 293–297. [7] J. Kawabe, Regularity and Lusin’s theorem for Riesz space-valued fuzzy measures, Fuzzy Sets Syst. 158 (2007) 895–903. [8] J. Li, Order continuous of monotone set function and convergence of measurable functions sequence, Appl. Math. Comput. 135 (2003) 211–218. [9] J. Li, R. Mesiar, Lusin’s theorem on monotone measure spaces, Fuzzy Sets Syst. 175 (2011) 75–86. [10] J. Li, M. Yasuda, Egoroff’s theorems on monotone non-additive measure space, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 12 (2004) 61–68. [11] J. Li, M. Yasuda, Lusin’s theorems on fuzzy measure spaces, Fuzzy Sets Syst. 146 (2004) 121–133. [12] T. Murofushi, K. Uchino, S. Asahina, Conditions for Egoroff’s theorem in non-additive measure theory, Fuzzy Sets Syst. 146 (2004) 135–146. [13] Q. Sun, Property (S) of fuzzy measure and Riesz’s theorem, Fuzzy Sets Syst. 62 (1994) 117–119. [14] Z. Wang, G.J. Klir, Fuzzy Measure Theory, Plenum Press, New York, 1992. [15] T. Watanabe, On sufficient conditions for the Egoroff theorem of an ordered topological vector space-valued non-additive measure, Fuzzy Sets Syst. 162 (2011) 79–83. [16] T. Watanabe, T. Kawasaki, T. Tanaka, On a sufficient condition of Lusin’s theorem for non-additive measures that take values in an ordered topological vector space, Fuzzy Sets Syst. 194 (2012) 66–75. [17] C. Wu, M. Ha, On the regularity of the fuzzy measure on metric fuzzy measure spaces, Fuzzy Sets Syst. 66 (1994) 373–379.