Note on the Regularity of Nonadditive Measures

The relations of continuity and regularity of nonadditive measures are considered in several papers [1–4]. In [5], Li et al. investigated the regularity in nonadditive measures. They proved that the null-additive fuzzy measures possess a Radon property (strong regularity) on a complete metric space. In [6], Kawabe also investigated the regularity in fuzzy measures taking value in Riesz spaces. He proved that every weakly null-additive Riesz space valued fuzzy measure on a complete or a locally compact, separable metric space is Radon, provided that the Riesz space has themultiple Egoroff property. On the other hand Li andMesiar [7] proved the regularity of nonadditive monotone measures. They proved that the equivalence condition of Egoroff ’s theorem implies regularity for the nonadditivemeasures by using pseudometric generating property of a set function. For information on real valued nonadditive measures, see [8–10]. In this paper, as notes, we prove that Egoroff ’s theorem implies Radon property (strong regularity) for nonadditive measures which have pseudometric generating property on a complete or a locally compact, separable metric space.


Introduction
The relations of continuity and regularity of nonadditive measures are considered in several papers [1][2][3][4]. In [5], Li et al. investigated the regularity in nonadditive measures. They proved that the null-additive fuzzy measures possess a Radon property (strong regularity) on a complete metric space. In [6], Kawabe also investigated the regularity in fuzzy measures taking value in Riesz spaces. He proved that every weakly null-additive Riesz space valued fuzzy measure on a complete or a locally compact, separable metric space is Radon, provided that the Riesz space has the multiple Egoroff property.
On the other hand Li and Mesiar [7] proved the regularity of nonadditive monotone measures. They proved that the equivalence condition of Egoroff 's theorem implies regularity for the nonadditive measures by using pseudometric generating property of a set function. For information on real valued nonadditive measures, see [8][9][10].
In this paper, as notes, we prove that Egoroff 's theorem implies Radon property (strong regularity) for nonadditive measures which have pseudometric generating property on a complete or a locally compact, separable metric space.

Preliminaries
Let be the set of real numbers and the set of natural numbers. In what follows, let ( , F) be a measurable space. Definition 1. A set function : F → is called a nonadditive measure if it satisfies the following two conditions: if , ∈ F and ⊂ , then ( ) ≤ ( ). Definition 2. Let : F → be a nonadditive measure.
(1) is said to be continuous from above if for any { } ⊂ F and ∈ F satisfying ↘ and there exists 0 with ( 0 ) < ∞ it holds that lim → ∞ ( ) = ( ). (2) is said to be continuous from below if for any { } ⊂ F and ∈ F satisfying ↗ it holds that lim → ∞ ( ) = ( ).
(3) is said to be fuzzy measure if it is continuous from above and below.
(9) is said to be autocontinuous if it is autocontinuous from above and below.
(1) A double sequence { , } ⊂ F is said to be aregulator if it satisfies the following two conditions:

Compact Measure and Regularity of Measure
In this section, we pick up several results for compact nonadditive measures and regularity of measures.
(1) A nonempty family K of subsets of is called a compact system if for any sequence [12].
(2) We say that is compact if there exists a compact system K such that for each ∈ F there are sequences { } ⊂ K and { } ⊂ F such that ⊂ ⊂ for all ∈ and lim → ∞ ( \ ) = 0.
The family of all compact subsets of a Hausdorff space is a compact system.
(2) The family of all finite unions of sets in a compact system is also compact [ In what follows, let ( , ) be a metric space. Denote by B( ) the -field of all Borel subsets of , that is, the -field generated by the open subsets of . A nonadditive measure defined on B( ) is called a nonadditive Borel measure on .

Proposition 9.
If satisfies pseudometric generating property, then it is weakly null-additive.
Proof. It is easy to see from the definition. For more information on regularity of nonadditive measures, see [5,6].

Radon Measure
In this section, as main results, if we assume that a nonadditive Borel measure satisfies the equivalence condition of Egoroff 's theorem and pseudometric generating property on a complete or a locally compact, separable metric space, then it is Radon. (2) is said to be tight if there is a sequence { } ∈ of compact sets such that lim → ∞ ( \ ) = 0. It is known that every finite Borel measure on a complete or a locally compact, separable metric space is Radon; see [16,Theorem 3.2] and [17, Theorems 6 and 9, Chapter II, Part I]. Its counterpart in nonadditive measure theory can be found in [5,9, Theorem 1, Lemma 2], which states that every Borel fuzzy measure on a complete separable metric space is tight, so that it is Radon if it is null-additive; see also [3,Theorem 2.3]. The following two theorems contain those previous results; see also [18,Theorem 12].

Theorem 17. Let be a complete separable metric space and
: B( ) → a nonadditive Borel measure on . If is weakly null-additive and satisfies the Egoroff condition, then it is tight. Moreover, if has pseudometric generating property and satisfies the Egoroff condition, then it is Radon.
To prove the theorem, we need the following; see [7,Proposition 3.7].

Proposition 18. Let : F → be a nonadditive measure. Then (i) implies (ii).
(i) is weakly null-additive and satisfies the Egoroff condition.
(ii) For each > 0 and double sequence Proof of Theorem 17. Since satisfies the Egoroff condition, by [19,Proposition 3], it is strongly order continuous. By Proposition 16 and Lemma 11, we have only to prove that is tight. Let { } ∈ be a countable dense subset of . For each , ∈ , denote by ( ) the closed ball with center and radius 1/ . For each , ∈ , put , := \ ∪ =1 ( ). Then, for any > 0 and ∈ , we have , ↘ 0, so that by Proposition 18, there exists a sequence { } of natural numbers such that Put := ∩ ∞ =1 ∪ =1 ( ). Then, each is closed and totally bounded, so that it is compact. Since \ = ∪ ∞ =1 , , it follows from (2)