数理解析研究所講究録
第 1820 巻 2012 年 43-56

43

THE L\’EVY-PROKHOROV TOPOLOGY ON NONADDITIVEMEASURES ON METRIC SPACES

信州大学工学部 河邊淳*(Jun Kawabe)

Faculty of Engineering, Shinshu University

ABSTRACT. We formalize the L\’evy-Prokhorov metric and the Fortet-Mourier
metric for nonadditive measures on a metric space and show that the L\’evy $top$ ology on every uniformly equi-autocontinuous set of Radon nonadditive measures
can be metrized by such metrics. This result is proved using the uniformity for
L\’evy convergence on a bounded subset of Lipschitz functions. We describe some
applications to stochastic convergence of a sequence of measurable mappings on a
nonadditive measure space.

1. INTRODUCTION
This is an announcement of the forthcoming paper [13]. Weak convergence ofmeasures on a topological space not only plays a very important role in probability
theory and statistics, but is also interesting from a topological measure theoretic
view, since it gives a convergence closely related to the topology of the space on

which the measures are defined. Thus, it is possible to study weak convergence ofmeasures on a topological space in association with some topological properties of thespace, such as the metrizability, separability and compactness; for comprehensive
information on this convergence, readers are referred to Alexandroff [1], Billingsley [2], Dudley [5], Parthasarathy [21], Vakhania et al. [23], Varadarajan [24], andreferences therein.
Nonadditive measures, which are set functions that are monotonic and vanish at
the empty set, have been extensively studied [4, 22, 25]. They are closely related to
nonadditive probability theory and the theory of capacities and random capacities.
Nonadditive measures have been used in expected utility theory, game theory, andsome economic topics under Knightian uncertainty [3, 8, 14, 16, 19, 20].
2010 Mathematics Subject Classification. Primary $28A33$ ; Secondary $28A12,28C15,28E10.$
Key words and phrases. Nonadditive measure; Weak convergence; L\’evy-Prokhorov metric,
Fortet-Mourier metric; L\’evy topology; Portmanteau theorem; Metrizability.
Research supported by Grant-in-Aid for Scientific Research (C) No. 23540192, Japan Society
for the Promotion of Sciences (JSPS).
$*$44
The notion of weak convergence of nonadditive measures was formulated by
Girotto and Holzer in a fairly abstract setting [10]. Some of their fundamental
results for weak convergence, such as the portmanteau theorem and the direct andconverse Prokhorov theorems, have been extended to the nonadditive case. In particular, the portmanteau theorem allows us to show that weak topology, which is
topology generated by weak convergence, coincides with the L\’evy topology, whichis the topology generated by convergence of measures on a special class of sets.
Although the metrizability of a topology is usually one of the main topics in topological theory, there seem to be no reports that focus on the metrizability of the
L\’evy topology of nonadditive measures. The aims of this paper are (i) to present
successful nonadditive analogs of the theory of weak convergence of measures witha particular focus on metrizability, and (ii) to supply weak convergence methods torelated fields.
The remainder of the paper is organized as follows. In Section 2 we recall the
notion of regularity systems of sets that are crucial for formalizing the portmanteau
theorem [10]. We also describe some new properties for later use and some examplesof regularity systems. In Section 3 we discuss the possibility of metrizing subsets
of nonadditive measures on a metric space. In particular, we show that the L\’evy
topology on the set of all Radon and autocontinuous nonadditive measures on a
separable metric space can be metrized. In Section 4 we introduce a Levy-Prokhorov
metric and a Fortet-Mourier metric on the space of nonadditive measures on ametric space. In Section 5 we show that the L\’evy topology on a uniformly equiautocontinuous set of Radon nonadditive measures can be metrized by the L\’evyProkhorov and Fortet-Mourier metrics. This result is proved using the uniformityof the L\’evy convergence on a bounded subset of Lipschitz functions. In Section 6
we describe some applications to stochastic convergence of a sequence of measurable
mappings defined on a nonadditive measure space.
2. REGULARITY SYSTEM AND THE PORTMANTEAU THEOREM
Throughout the paper, unless stated otherwise, $X$ is a Hausdorff space andbe the family of all subsets of
is a field containing all open subsets of $X$ . Let
denote the closure, interior, and boundary, andX. For a set $A\subset X,$
and denote the set of all real numbers and the set of allof , respectively.
be the characteristic function of a set . For
natural numbers, respectively. Let$:=\chi_{A}(x)$ for all
denotes the unit mass at $x\in X$ defined byeach $x\in X,$$A\subset X$ . We say that a set function
is a nonadditive measure on:$\mathcal{B}$$2^{X}$$A^{-},$$A$$\mathbb{R}$$A^{o}$$\partial A$$\mathbb{N}$$A$$\chi_{A}$$\delta_{x}(A)$$\delta_{x}$$\mu$$\mathcal{B}arrow[0, \infty)$

45
if
and $\mu(A)\leq\mu(B)$ whenever $A,$
and $A\subset B$ . Letbethe set of all such measures. As usual, the conjugate of is defined by$:=$$\mu(X)-\mu(B^{c})$ for all, wheredenotes the complement of the set . For asubset, let; this is called the conjugate space ofGirotto and Holzer formulated a nonadditive extension of the portmanteau theorem
for weak convergence of measures using the following regularity systems of sets [10].
$X$$\mu(\emptyset)=0$$B\in \mathcal{B}$$\mathcal{M}(X)$$\overline{\mu}$$\overline{\mathcal{P}}$$B$$B^{c}$$B\in \mathcal{B}$$\mathcal{P}\subset \mathcal{M}(X)$$\overline{\mu}(B)$$\mu$$:=\{\overline{\mu}:\mu\in \mathcal{P}\}$$\mathcal{P}.$Definition 1. Let.
(1) The outer regularization$:=$of is the nonadditive measure defined by{ $\mu(U)$ : $A\subset U$ and is open} for every subset of $X$ , and the inner regular$\mu\in \mathcal{M}(X)$$\mu^{*}$$\inf$$\mu^{*}(A)$

$\mu$

$A$

$U$

$:= \sup\{\mu(C)$ :is the nonadditive measure defined byis closed} for every subset
of $X$ . We denote by
the family of all
satisfying $\mu(B)=\mu^{*}(B)=\mu_{*}(B)$ and we call this the -regularity system.
(2) The strongly outer regularizationof is the nonadditive measure defined$:= \inf$ $\mu(C)$ : $A\subset C$ andbyis closed} for every subset
of $X,${and the strongly inner regularizationof is the nonadditive measure defined by$:= \sup$ { $\mu(U)$ : $U\subset A$ andis open} for every subset
of $X$ . We
denote by
the family of all
satisfying
and we callthis the -strong regularity system.

izationand$A$$\mu_{*}$

of$C\subset$$\mu_{*}(A)$

$\mu$

$C$

$A$

$\mathcal{R}_{\mu}$$B\in \mathcal{B}$$\mu$$\mu^{\natural}$$\mu$$\mu^{\natural}(A)$$A$$C\in \mathcal{R}_{\mu}$$\mu$$\mu_{\natural}$$\mu_{\natural}(A)$$A$$U\in \mathcal{R}_{\mu}$$B\in \mathcal{B}$$\mathcal{R}_{\mathring{\mu}}$$\mu(B)=\mu^{\natural}(B)=\mu_{\natural}(B)$$\mu$Remark 1. (1) This type of regularity was introduced and discussed by Narukawa
and Murofushi for nonadditive measures on locally compact spaces [18].
(2) The regularity notion defined above is different from that in [12, Definition 5]
and it is appropriate for our purpose. If is autocontinuous, then the former follows
from the latter.
$\mu$

For later use, we collect some properties of the -regularity and -strong regularity
systems; see [10, Remark 2.2 and Theorem 2.3] for proofs.
$\mu$

$\mu$

Proposition 1. Let
(1)
(2)
(3)
(4)
Let
for allby$\mu\in \mathcal{M}(X)$and$B\in \mathcal{B}.$$\mathcal{R}_{\mu}^{o}\subset \mathcal{R}_{\mu}.$$\emptyset,$$X\in \mathcal{R}_{\mu}^{o}.$$B\in \mathcal{R}_{\mu}$$B\in \mathcal{R}_{\mu}^{o}$if and only if
if and only if$B^{c} \in \mathcal{R}\frac{\circ}{\mu}.$and let be a real-valued function on $X$ such that $\{f>t\}\in \mathcal{B}$. The (asymmetric) Choquet integral of with respect to is defined$\mu\in \mathcal{M}(X)$$t\in \mathbb{R}$$B^{c}\in \mathcal{R}_{\overline{\mu}}.$

$f$

$f$

$\int_{X}fd\mu:=\int_{0}^{\infty}\mu(\{f>t\})dt-\int_{-\infty}^{0}\{\mu(X)-\mu(\{f>t\})\}dt$

$\mu$

46
whenever the Lebesgue integrals of the right-hand side of the above equation are notboth . We say that is Choquet integrable if $\int_{X}fd\mu<\infty$ . Every bounded continuous function and every characteristic function of a set in is Choquet integrable.
always denotes the Choquet integral.
In this paper, the symbol
We denote by $C_{b}(X)$ the real Banach space of all bounded, continuous real-valued$:= \sup_{x\in X}|f(x)|$ . The weak topology on
isfunctions on $X$ with norm
, the basic neighborhoods of are sets ofthe topology such that, for any$f$$\infty$$\mathcal{B}$$\int_{X}fd\mu$$\mathcal{M}(X)$$\Vert f\Vert_{\infty}$$\mu\in \mathcal{M}(X)$$\mu$the form:$V_{\mu,\epsilon,f_{1},\ldots,f_{k}}:= \{\nu\in \mathcal{M}(X):|\int_{X}f_{i}dv-\int_{X}f_{i}d\mu|<\epsilon(i=1, \ldots, k)\},$$f_{k}\in C_{b}(X)$ [ $10$ , Definition 3.1]. Given a net
, and
where
weakly converges to and write, we say thatinand
if andconverges to with respect to the weak topology. Obviously,
iffor every $f\in C_{b}(X)$ .
only if
There are already some convergence notions of nonadditive measures [17, 26].
However, the following proposition shows that weak convergence has the advantage
of giving a convergence related to the topology of $X$ ; see [24, Theorem II.9] for the$\epsilon>0,$$k\in \mathbb{N}$$\{\mu_{\alpha}\}_{\alpha\in\Gamma}$$f_{1},$$\ldots,$$\mu\in \mathcal{M}(X)$$\mathcal{M}(X)$$\mu_{\alpha}$$\mu$$\mu_{\alpha}arrow^{w}\mu$$\mu_{\alpha}arrow^{w}\mu$$\mu$$\mu_{\alpha}$$\int_{X}fd\mu_{\alpha}arrow\int_{X}fd\mu$

proof.

Proposition 2. Assume that $\mathcal{D}(X);=\{\delta_{x}:x\in X\}$ is endowed with the relative. Then $X$ is homeomorphic totopology induced by the weak topology on
if and only if $X$ is completely regular.
$\mathcal{D}(X)$$\mathcal{M}(X)$Now we are ready to introduce a nonadditive extension of the portmanteau theorem, which gives a comprehensive list of conditions equivalent to weak convergence.
The following is a special case of [10, Theorem 3.7] and is enough for our purposein this paper.

Theorem 1 (The portmanteau theorem). Let $X$ be a normal space and let. The following conditions are equivalent:andbe a net in$\{\mu_{\alpha}\}_{\alpha\in\Gamma}$$\mathcal{M}(X)$(i)$\mu_{\alpha}arrow^{w}\mu.$(ii)$\overline{\mu}_{\alpha}arrow^{w}\overline{\mu}.$(iii)$\mu\in \mathcal{M}(X)$$\lim\sup_{\alpha\in\Gamma}\mu_{\alpha}(C)\leq\mu(C)$$\mathcal{R}_{\mu}$and every openand$\mu(U)\leq\lim\inf_{\alpha\in\Gamma}\mu_{\alpha}(U)$$U\in \mathcal{R}_{\mu}.$(iv) $\mu_{\alpha}(B)arrow\mu(B)$ for every$B\in \mathcal{R}_{\mu}^{o}.$(v) $\mu_{\alpha}(U)arrow\mu(U)$ for every open

$U \in \mathcal{R}_{\mu}^{o}\cap \mathcal{R}\frac{\circ}{\mu}.$(vi) $\mu_{\alpha}(C)arrow\mu(C)$ for every closed$C \in \mathcal{R}_{\mu}^{o}\cap \mathcal{R}\frac{\circ}{\mu}.$for every closed$C\in$

47
Recall that the L\’evy topology onis the topology such that, for any, the basic neighborhoods of are the sets of the form:$\mathcal{M}(X)$$\mathcal{M}(X)$$\mu\in$$\mu$$W_{\mu,\epsilon,B_{1},\ldots,B_{k}}:= \{\nu\in \mathcal{M}(X):|^{\frac{v}{\nu}}(B_{i})-\frac{\mu}{\mu}(B_{i})||(B_{i})-(B_{i})|<\epsilon<\epsilon (i=1,2, \ldots, k)\},$

where$\epsilon>0,$$\{\mu_{\alpha}\}_{\alpha\in\Gamma}$in
if$k\in \mathbb{N}$$\mathcal{M}(X)$

, and

and

$B_{1},$$\mu_{\alpha}$$B_{k} \in \mathcal{R}_{\mu}^{o}\cap \mathcal{R}\frac{\circ}{\mu}$$\mu\in \mathcal{M}(X)$converges totheorem shows that$\mu_{\alpha}arrow^{L}\mu$[ $10$ , Definition 4.1]. Given a net, we say thatL\’evy converges to and writewith respect to the L\’evy topology. The portmanteau$\ldots,$$\mu$$\mu_{\alpha}$$\mu$

if and only if $\mu_{\alpha}(B)arrow\mu(B)$ for every, andhence the weak topology and the L\’evy topology coincide on.
In the rest of this section, we study regularity systems in more detail. To thisend, we introduce the following continuity notion of nonadditive measures.
$\mu_{\alpha}arrow^{L}\mu$$B\in \mathcal{R}_{\mu}^{o}$$\mathcal{M}(X)$Definition 2. Let.
(1) is said to be -continuous if
wheneverdecreasing sequence of closed sets with
(2) is said to be -continuous if
wheneverincreasing sequence of open sets with
(3) is said to be -continuous if it is -continuous and -continuous.
$\mu\in \mathcal{M}(X)$$\mu(C)=\inf_{n\in \mathbb{N}}\mu(C_{n})$

$c$

$\mu$

$\{C_{n}\}_{n\in \mathbb{N}}$is a$C= \bigcap_{n=1}^{\infty}C_{n}.$$\mu(U)=\sup_{n\in \mathbb{N}}\mu(U_{n})$

$0$

$\mu$

$\{U_{n}\}_{n\in \mathbb{N}}$is an$U= \bigcup_{n=1}^{\infty}U_{n}.$

$co$

$\mu$

$c$

$0$

Remark 2. (1) This type of continuity was introduced and discussed by Narukawa
and Murofushi for nonadditive measures on locally compact spaces [18].
(2) The generalized sequence versions of -continuity and -continuity are called
the total -continuity and total -continuity, respectively. They have previously been
discussed with applications to convergence theorems for Choquet integrals [12].
$c$

$c$

$0$

$0$

In general, -continuity and -continuity are independent of each other (Examples 2 and 3). We say that $X$ is perfectly normal if $X$ is normal and every closed$c$subset of $X$ is a$0$

$G_{\delta}$-set, which is a countable intersection of open sets. Every metricspace is perfectly normal [6, Corollary 4.1.13].
Proposition 3. Let $X$ be perfectly normal and let.
(1) If is -continuous, thencontains all closed subsets of $X.$
(2) If is -continuous, thencontains all open subsets of $X.$$\mu\in \mathcal{M}(X)$

$\mu$

$c$

$\mathcal{R}_{\mu}$

$\mu$

$0$

$\mathcal{R}_{\mu}$

If is additive, that is, $\mu(A\cup B)=\mu(A)+\mu(B)$ whenever $A,$andthen the -regularity systemand the -strong regularity system
are fields,$A\cap B=\emptyset,$$B\in \mathcal{B}$

$\mu$

$\mu$

$\mathcal{R}_{\mu}$$\mu$$\mathcal{R}_{\mathring{\mu}}$andis the family of allwith
and $\mu(\partial B)=0$ ; see [9, Theorems 2.2
and 2.5] and [10, Remark 2.2(ii)], but this is not the case for nonadditive measures(Examples 1 and 2).
$\mathcal{R}_{\mu}^{o}$$B\in \mathcal{B}$$\partial B\in \mathcal{R}_{\mu}$48
Proposition 4. Let(1) If$B\in \mathcal{R}_{\mu}^{o}$$\mu\in \mathcal{M}(X)$and$B\in \mathcal{B}.$, then $\mu(B^{-})=\mu(B^{o})$ .

(2) Let $X$ be perfectly normal. Assume that$\mu$

is $co$ -continuous. Then

$B\in \mathcal{R}_{\mu}^{o}$if

and only if $\mu(B^{-})=\mu(B^{o})$ .
with $\mu(B^{-})=\mu(B^{o})$ thus $\mu(\partial B)=0)$ is
In probability theory, a Borel set
called a -continuity set [2, page 15]. Therefore, Proposition 4 gives a sufficientwith the set of allcondition for coincidence of the -strong regularity system
$B$

$($

$\mu$

$\mathcal{R}_{\mu}^{o}$

$\mu$

$\mu$

. Note that, in general, $\mu(\partial B)=0$ is not equivalent to $\mu(B^{-})=$-continuity(Examples 1 and 2).
$B\in \mathcal{R}_{\mu}$$\mu(B^{o})$Example 1. Let $X$ $:=[0,1]$ . We define the nonadditive measure$\mu$: $2^{X}arrow[0,1]$ as$\mu(A):=\{\begin{array}{l}0 if A\neq X1 if A=X\end{array}$for each subset of $X.$
(1) is co-continuous.
(2) Let $:=(0,1)$ . Then
$A$

$\mu$

$U$

(3) Let$U$$:=(0,1)$ .

$\mu(U^{-})\neq\mu(U^{o})$Then$U^{c}\in \mathcal{R}_{\mu}^{o}$. Thus,, but
and $\mu(\partial U)=0$ , but$U\not\in \mathcal{R}_{\mu}^{o}$$\mathcal{R}_{\mu}^{o}$$\partial U\in \mathcal{R}_{\mu}$

is not a field.
. Furthermore,$U\not\in \mathcal{R}_{\mu}^{o}$.

Example 2. Let $X$ $:=[0,1]$ . We define the nonadditive measure$\mu$: $2^{X}arrow[0,1]$ as$\mu(A):=\{\begin{array}{l}0 if A\subset\{0,1\}1 if A\not\subset\{0,1\}\end{array}$for each subset of $X.$
(1) is -continuous but is not -continuous.
, but(2) Let $:=(0,1)$ . Then
, but$:=(0,1/2)$ . Then
(3) Let
$A$

$\mu$

$c$

$0$

$U$

$U\in \mathcal{R}_{\mu}$$V$$\mu(V^{o})$$V\in \mathcal{R}_{\mu}^{o}$$U^{c}\not\in \mathcal{R}_{\mu}$. Thus,$\mu(\partial V)\neq 0$$\mathcal{R}_{\mu}$

is not a field.

. Furthermore, $\mu(V^{-})=$.

Example 3. Let $X$ $:=(0,1)$ . We define the nonadditive measure$\mu$: $2^{X}arrow[0,1]$ as$\mu(A):=\{\begin{array}{l}0 if A\neq X1 if A=X\end{array}$for each subset$A$of $X$ . Then$\mu$

is -continuous but is not -continuous.
$c$

3. METRIZATION OF SUBSETS OF$\mathcal{M}(X)$in the caseIn this section, we discuss the possibility of metrizing subsets ofin which $X$ is a separable metric space. In the remainder of the paper, let ( $X$ , d) bea metric space with metric . Recall that a real-valued function on $X$ is called$\mathcal{M}(X)$

$d$

$f$

49
Lipschitz if there is a constant $K>0$ such that $|f(x)-f(y)|\leq Kd(x, y)$ for all. The Lipschitz seminorm is defined as $\Vert f\Vert_{L}:=\sup\{|f(x)-f(y)|/d(x, y)$ :$y\in X,$ $x\neq y\}$ . Let $BL(X, d)$ denote the real Banach space of all bounded, realvalued Lipschitz functions on $X$ with norm
. For any subset
of $X$ , let $d(x, A)$ be the distance from to , defined as $d(x, A)$ $:= \inf\{d(x, a)$ :$x,$ $y\in

X$

$x,$

$f$

$\Vert f\Vert_{BL}$

$A$

$x$

$:=\Vert f\Vert_{L}+\Vert f\Vert_{\infty}$$A$$a\in A\}.$

Proposition 5. Let$BL(X, d)$ .

(1) If(2) If

$\mu$

$\mu$

and

$v$

and

$v$

$v\in \mathcal{M}(X)$. Assume that $\int_{X}fd\mu=\int_{X}fdv$ for every $f\in$are -continuou , then $\mu(C)=v(C)$ for every closed
are -continuous, then $\mu(U)=v(U)$ for every open
$c$

$s$

$C\subset X.$$U\subset X.$$0$

Proposition 6. Let$co$$\mu,$$\{\mu_{\alpha}\}_{\alpha\in\Gamma}$

be a net in$\mathcal{M}(X)$and$\mu\in \mathcal{M}(X)$. Assume that

$\mu$

is

-continuous. The following conditions are equivalent:(i)$\mu_{\alpha}arrow^{w}\mu.$$( ii)\int_{X}fd\mu_{\alpha}arrow\int_{X}fd\mu$Letfor every $f\in BL(X, d)$ .

denote the space of allthat are co-continuous and$\mu(B)=\mu^{*}(B)=\mu_{*}(B)$ for all. Then it is readily evident thatcoincides with its conjugate space. The following theorem can be provedby Theorem 1 and Propositions 2, 5 and 6 in the same way as [21, Theorem II.6.2].
$\mathcal{M}_{rco}(X)$$\mu\in \mathcal{M}(X)$$SatiS\mathfrak{h}r$$B\in \mathcal{B}$$\mathcal{M}_{r\omega}(X)$$\overline{\mathcal{M}_{rco}(X)}$

Theorem 2. The L\’evy topology on
is metrizable as a sepamble metricspace if and only if $X$ is a sepamble metric space.
$\mathcal{M}_{rco}(X)$

Let $\mu\in \mathcal{M}(X)$ .

We say thatis autocontinuous if $\lim_{narrow\infty}\mu(A\cup B_{n})=$whenever $A,$
for alland $\lim_{narrow\infty}\mu(B_{n})=0.$
We also say that is Radon if, for every, there are an increasing sequenceof compact sets and a decreasing sequenceof open sets such thatfor alland$\mu$$\lim_{narrow\infty}\mu(A\backslash B_{n})=\mu(A)$$B_{n}\in \mathcal{B}$$B\in \mathcal{B}$$\mu$$\{K_{n}\}_{n\in \mathbb{N}}$$\{U_{n}\}_{n\in \mathbb{N}}$$K_{n}\subset B\subset U_{n}$$n\in \mathbb{N}$

Proposition 7. Let$\mu\in \mathcal{M}(X)$$\mathcal{M}_{rco}(X)$$n\in \mathbb{N}$$\lim_{narrow\infty}\mu(U_{n}\backslash K_{n})=0.$

.

If

$\mu$

is Radon and autocontinuous, then$\mu,\overline{\mu}\in$.

By Theorem 2 and Proposition 7, we have the following theorem.
Theorem 3. Each of the L\’evy topologies on the set of all Radon and autocontinuousand on its conjugate space is metrizable as a sepamble metric space if
and only if $X$ is a sepamble metric space.

$\mu\in \mathcal{M}(X)$50
4. THE L\’E$VY$-PROKHOROV METRIC AND THE $FoRTET-$ MOURIER METRIC
In this section, we introduce a L\’evy-Prokhorov metric and a Fortet-Mourier$:=\{x\in X:d(x, A)<\epsilon\}.$, let. For any subset of $X$ andmetric on
Definition 3. For any$\epsilon$

for all$B\in \mathcal{B}$$\epsilon>0$$A$$\mathcal{M}(X)$$v\in \mathcal{M}(X)$$\mu,$, let$\rho(\mu, v)$$A^{\epsilon}$$:=iryf\{\epsilon>0$: $\mu(B)\leq v(B^{\epsilon})+$}.

In the same way as for the additive case [5, Theorem 11.3.1], it can be proved
for anythat satisfies $\rho(\mu, v)\geq 0,$ $\rho(\mu, \mu)=0$ , and$\rho(\mu, \nu)\leq\rho(\mu, \lambda)+\rho(\lambda, \nu)$$\rho$$\mu,$$v,$ $\lambda\in \mathcal{M}(X)$.

Example 4. The symmetry relation $\rho(\mu, \nu)=\rho(v,\mu)$ does not hold in general. Let$X$ $:=[0,1]$ and let. We define the nonadditive measure $v:2^{X}arrow[0,1]$ as$\mu$$:=\delta_{0}$$\nu(A):=\{\begin{array}{ll}0 if A=\emptyset 1/2 if A\neq\emptyset and 0\not\in A1 if0\in A.\end{array}$

Then $\rho(\mu, \nu)=0$ , but $\rho(v,\mu)=1/2.$
Proposition 8. Let.

$\mu,$$v\in \mathcal{M}(X)$and assume that $\mu(X)=v(X)$ . Then $\rho(\mu, v)=$$\rho(\overline{\nu},\overline{\mu})$

Example 5. Proposition 8 is no longer true if $\mu(X)\neq\nu(X)$ . Let $X$ $:=[0,1]$ andand be the same as in Example 4. Then $\mu(X)=1/2\neq 1=v(X)$ .
let
$\mu$

$v$

$:=\overline{\delta}_{0}/2$

The conjugate$\overline{\nu}$

of

is

$v$

$\overline{v}(A):=\{\begin{array}{ll}0 if A=\emptyset or 0\not\in A1/2 if A\neq X and 0\in A1 if A=X.\end{array}$

Then $\rho(\mu, v)=0$ , but$\rho(\overline{v},\overline{\mu})=1/2.$$v\in \mathcal{M}(X)$ , letDefinition 4. For any; we call this the L\’evy-Prokhorov semimetric.
$\pi(\mu, \nu)$$\mu,$$:=\rho(\mu, \nu)+\rho(v, \mu)+\rho(\overline{\mu},\overline{v})+$$\rho(\overline{\nu},\overline{\mu})$

The following properties can be proved by the standard argument in the additivecase.
.
Proposition 9. Let.
. Furthermore,(1) is a semimetric on(2) Assume that and are -continuous and $\mu(B)=\mu_{*}(B)$ and $\nu(B)=v_{*}(B)$. Then $\mu=\nu$ whenever $\rho(\mu, \nu)=\rho(v, \mu)=0.$
for all(3) Assume that and are -continuous and $\mu(B)=\mu^{*}(B)$ and $\nu(B)=v^{*}(B)$$\mu,$$\nu\in \mathcal{M}(X)$$\mathcal{M}(X)$

$\pi$

$\mu$

$\nu$

$c$

$\mu$

$\nu$

$0$

$\pi(\mu, \nu)=\pi(\overline{\mu},\overline{\nu})$$B\in \mathcal{B}$for all$B\in \mathcal{B}$. Then $\mu=\nu$ whenever$\rho(\overline{\mu},\overline{\nu})=\rho(\overline{v},\overline{\mu})=0.$

51
Thus, by Propositions 7 and 9, we have the following.

Proposition 10. The L\’evy-Prokhorov semimetric$\pi$

is a metric onhence it is also a metric on the set of all Radon and autocontinuous
on its conjugate space.
Next we define the Fortet-Mourier metric onconvergence [7].
Definition 5. For any$\kappa(\mu, v)$$v\in \mathcal{M}(X)$$\mu,$$\mathcal{M}(X)$$\mathcal{M}_{r\omega}(X)$, and$\mu\in \mathcal{M}(X)$andthat is closely linked to weak, let$:= \sup\{|\int_{X}fd\mu-\int_{X}fd\nu|$: $f\in BL(X, d),$$\Vert f\Vert_{BL}\leq 1\}.$

We call this the $Fortet-$Mourier semimetric.
Proposition 11. Let.
(1) is a semimetric on. Furthermore,.
(2) Assume that and are -continuous and $\mu(B)=\mu_{*}(B)$ and $v(B)=v_{*}(B)$. Then $\mu=\nu$ whenever $\kappa(\mu, v)=0.$
for all(3) Assume that and are -continuous and $\mu(B)=\mu^{*}(B)$ and $\nu(B)=v^{*}(B)$. Then $\mu=v$ whenever $\kappa(\mu, v)=0.$
for all(4) The semimetric is a metric on, and hence it is also a metric on
the set of all Radon and autocontinuousand on its conjugate$\mu,$$v\in \mathcal{M}(X)$$\mathcal{M}(X)$$\kappa$

$\mu$

$v$

$c$

$\nu$

$0$

$\kappa(\mu, \nu)=\kappa(\overline{\mu},\overline{v})$$B\in \mathcal{B}$$\mu$$B\in \mathcal{B}$$\mathcal{M}_{rco}(X)$$\kappa$$\mu\in \mathcal{M}(X)$

Proposition 12. For any$\mu,$$v\in \mathcal{M}(X)$$\mathcal{S}pace.$, we have $\rho(\mu, v)\leq\kappa(\mu, v)+\kappa(\mu, v)^{1/2}.$

5. METRIZATION BY THE $LEVY$’-PROKHOROV AND $FoRTET-$ MOURIER METRICS
In this section, we show that the L\’evy topology on certain sets of nonadditive

measures can be metrized by the L\’evy-Prokhorov metric and the Fortet-Mourier
metric. To this end, we introduce the notion of uniform equi-autocontinuity of

nonadditive measures.
Definition 6. Let be a subset of.
and(1) is said to be uniformly autocontinuous if, for every, there is $\delta>0$$\mu(B)<\delta$ implies $\mu(A\cup B)-\epsilon\leq\mu(A)\leq\mu(A\backslash B)+\epsilon.$such that, for any $A,$(2) is said to be uniformly equi-autocontinuous if, for every, there is $\delta>0$$\mu(B)<\delta$ implies $\mu(A\cup B)-\epsilon\leq\mu(A)\leq$such that, for any
and $A,$$\mathcal{M}(X)$$\mathcal{P}$$\mu\in \mathcal{M}(X)$$\epsilon>0$$\mu$$B\in \mathcal{B},$$\epsilon>0$$\mathcal{P}$$B\in \mathcal{B},$$\mu\in \mathcal{P}$$\mu(A\backslash B)+\epsilon.$Let. We say that is subadditive if $\mu(A\cup B)\leq\mu(A)+\mu(B)$ for every. Obviously, every subadditive measure is uniformly autocontinuous, and$\mu\in \mathcal{M}(X)$$A,$ $B\in \mathcal{B}$

$\mu$

52
every subset of the subadditive measures is uniformly equi-autocontinuous. Recall, there is a compact subset $K$ of $X$ such thatis tight if, for every
thatis tight.
. It is obvious that every Radon
Unlike the Lebesgue integral, the Choquet integral is generally nonlinear withrespect to its integrand owing to the nonadditivity of . That is, we may have$g)d \mu\neq\int_{X}fd\mu+\int_{X}gd\mu$ for some functions
and . This is one of the reasons why
we should assume the uniform equi-autocontinuity of a set of nonadditive measuresin the following theorems. The following theorem shows the uniformity for the L\’evy
convergence on a bounded subset of Lipschitz functions and it plays an essential
part in the proof of Theorem 5.
$\epsilon>0$$\mu$$\mu\in \mathcal{M}(X)$$\mu(X\backslash K)<\epsilon$$\int_{X}(f+$

$\mu$

$f$

$g$

. Assume thatandbe a net in
Theorem 4. Let
is unifomly equi-autocontinuous and is uniformly autocontinuous. Furthermore,assume that is tight. The following conditions are equivalent:$\mu\in \mathcal{M}(X)$$\mathcal{M}(X)$$\{\mu_{\alpha}\}_{\alpha\in\Gamma}$$\{\mu_{\alpha}\}_{\alpha\in\Gamma}$

$\mu$

$\mu$

(i)
(ii)

$\int_{X}fd\mu_{\alpha}arrow\int_{X}fd\mu$

for every $f\in BL(X, d)$ .

$\kappa(\mu_{\alpha}, \mu)arrow 0.$. Assume thatandbe a net in
Theorem 5. Let
is uniformly equi-autocontinuous and is uniformly autocontinuous. Furthemore,assume that is tight and $co$ -continuous. The following conditions are equivalent:$\mu\in \mathcal{M}(X)$$\mathcal{M}(X)$$\{\mu_{\alpha}\}_{\alpha\in\Gamma}$$\{\mu_{\alpha}\}_{\alpha\in\Gamma}$

$\mu$

$\mu$

(i)
(ii)
(iii)
(iv)
(v)

$\mu_{\alpha}arrow^{L}\mu.$$\mu_{\alpha}arrow^{w}\mu.$$\int_{X}fd\mu_{\alpha}arrow\int_{X}fd\mu$

for every $f\in BL(X, d)$ .

$\kappa(\mu_{\alpha}, \mu)arrow 0.$$\pi(\mu_{\alpha}, \mu)arrow 0.$

Remark 3. In Theorem 5, the co-continuity of.
implication$\mu$

is needed only in the proof of the$(v)\Rightarrow(i)$

and
Since
proposition easily follows from Theorem 1.
$\pi(\mu, \nu)=\pi(\overline{\mu},\overline{\nu})$$\kappa(\mu, \nu)=\kappa(\overline{\mu},\overline{v})$for any$\mu,$$\nu\in \mathcal{M}(X)$, the following

is metrizable with. The L\’evy topology on
Proposition 13. Let
respect to (or ) if and only if the L\’evy topology on is metrizable with respect
to (or ).
$\mathcal{P}$$\mathcal{P}\subset \mathcal{M}(X)$$\overline{\mathcal{P}}$

$\pi$

$\pi$

$\kappa$$\kappa$

By Theorem 5 and Propositions 13, we have the following corollaries.

53
Corollary 1. Let
be uniformly equi-autocontinuous. Assume that every
is Radon. Each L\’evy topology on and on its conjugate space is metrizablewith respect to and$\mathcal{P}\subset \mathcal{M}(X)$$\mu\in \mathcal{P}$$\overline{\mathcal{P}}$$\mathcal{P}$$\pi$$\kappa.$

Corollary 2. Each L\’evy topology on the set of all Radon and subadditive

and on its conjugate space is metrizable with respect to

$\pi$

and

$\mu\in \mathcal{M}(X)$$\kappa.$

Let. We say that is submodular if $\mu(A\cup B)+\mu(A\cap B)\leq\mu(A)+\mu(B)$for every $A,$and it is supermodular if the reverse inequality holds. We also
say thatis continuous if it is continuous from above, that is, $\mu(A_{n})\downarrow\mu(A)$$\mu\in \mathcal{M}(X)$$\mu$$B\in \mathcal{B}$$\mu$wheneversatisfy, and it is continuous from below,
andthat is, $\mu(A_{n})\uparrow\mu(A)$ , wheneverandsatisfy. We
denote by
the set of all continuous and submodular
and$A\in \mathcal{B}$$\{A_{n}\}_{n\in \mathbb{N}}\subset \mathcal{B}$$A_{n}\downarrow A$$A\in \mathcal{B}$$\{A_{n}\}_{n\in \mathbb{N}}\subset \mathcal{B}$$A_{n}\uparrow A$$\mu\in \mathcal{M}(X)$$CS\mathcal{U}\mathcal{B}\mathcal{M}(X)$by$\mathcal{C}S\mathcal{U}\mathcal{P}\mathcal{M}(X)$the set of all continuous and supermodular$\mu\in \mathcal{M}(X)$. Let$\mathcal{B}(X)$be the -field of all Borel subsets of $X$ , that is, the -field generated by the open
subsets of $X.$$\sigma$$\sigma$

Theorem 6. Let $X$ be a complete or locally compact, sepamble metric space. Let$\mathcal{B}=\mathcal{B}(X)$(1).

$\overline{CS\mathcal{U}\mathcal{B}\mathcal{M}(X)}=CS\mathcal{U}\mathcal{P}\mathcal{M}(X)$(2) The L\’evy topology on(3) The L\’evy topology on

Letand let.

$\mathcal{C}\mathcal{S}\mathcal{U}\mathcal{B}\mathcal{M}(X)$$CS\mathcal{U}\mathcal{P}\mathcal{M}(X)$is metrizable with respect tois metrizable with respect to. We say that

$\pi$

$\pi$

and
and

$\kappa.$$\kappa.$satisfies the -rule if $\mu(A\cup B)=$$\mu(A)+\mu(B)+\lambda\cdot\mu(A)\cdot\mu(B)$ whenever $A,$
[ $25$ , Definition 4.3].
andEvery nonadditive measure satisfying the -rule is subadditive when $\lambda<0$ ; it issuperadditive when $\lambda>0$ ; and it is additive when $\lambda=0$ . The following example
gives a uniformly equi-autocontinuous set of Radon nonadditive measures that areboth subadditive and superadditive.
$\mu\in \mathcal{M}(X)$$\lambda\in \mathbb{R}$$\lambda$$\mu$$B\in \mathcal{B}$$A\cap B=\emptyset$$\lambda$

Example 6. Let. Let$\mathcal{Q}$be a set of Radon finitely additive

andbe real constants withthe function: $[0, \infty)arrow[0, \infty)$ as$\infty$$\lambda_{1}$$\lambda_{2}$$v\in \mathcal{M}(X)$$\lambda_{1}<0<\lambda_{2}$with $\sup_{\nu\in Q}\nu(X)<$. For each$\lambda\in \mathbb{R}$, we define$\varphi_{\lambda}$$\varphi_{\lambda}(t):=\{\begin{array}{ll}\frac{e^{\lambda t}-1}{\lambda} (\lambda\neq 0)t (\lambda=0)\end{array}$$:=$and we define the Radon nonadditive measureasfor all. Then it is routine to show that eachsatisfies the -ruleand the set
is uniformly equi-autocontinuous.
$\varphi_{\lambda}\circ v:\mathcal{B}arrow[0, \infty)$$B\in \mathcal{B}$$\varphi_{\lambda}(\nu(B))$$\mathcal{P}$$:=\{\varphi_{\lambda}ov :$(\varphi_{\lambda}\circ v)(B)$$\lambda$$\varphi_{\lambda}ov$\lambda_{1}\leq\lambda\leq\lambda_{2}, v\in \mathcal{Q}\}$

54
Consequently, by Corollary 1, the L\’evy topology on
$\pi$

and

$\mathcal{P}$is metrizable with respect to$\kappa.$, there is a
is said to be uniformly tight if, for every
A set
for all $\mu\in M$ . The uniform
compact subset $K$ of $X$ such thatequi-autocontinuity can also be utilized in the proof of a nonadditive version of thewell-known LeCam theorem concerning the uniform tightness of a weakly convergent$\epsilon>0$$M\subset \mathcal{M}(X)$$\mu(X\backslash K)<\epsilon$

sequence of measures [5, Theorem 11.5.3].

. Assume thatbe a sequence and let
is Radon. Furthermore, assume
is uniformly equi-autocontinuous and each, thenis uniformly tight.
that is tight and -continuous. If

Theorem 7. Let$\mu\in \mathcal{M}(X)$$\{\mu_{n}\}_{n\in N}\subset \mathcal{M}(X)$$\{\mu_{n}\}$$\mu_{n}$$\mu_{n}arrow^{L}\mu$

$c$

$\mu$

$\{\mu_{n}\}$6. APPLICATIONSIn this section, we give some applications to stochastic convergence of sequencesof metric space-valued measurable mappings on a nonadditive measure space. Let
is a -field of
is a non-empty set andbe a measurable space, that is,.
subsets of . Throughout this section, let ( $X$ , d) be a metric space and letthat are measurable withthe set of all mappings :We denote by$\Omega$$(\Omega, \mathcal{A})$$\mathcal{A}$$\sigma$$\mathcal{B}=\mathcal{B}(X)$$\Omega$$\mathcal{F}(\Omega, X)$$\xi$$\Omegaarrow X$.
respect to the -fields andbe a nonadditive measure. The nonadditive
. Let
Let
measure $P\circ\xi^{-1}\in \mathcal{M}(X)$ defined by $P\circ\xi^{-1}(E)$ $:=P(\xi^{-1}(E))$ for every $E\in$, one of which is separably
is called the distribution of . Letvalued, that is, the range space is a separable subset of $X$ . Then the function$\mathcal{A}$$\sigma$$\mathcal{B}(X)$$P:\mathcal{A}arrow[0, \infty)$$\xi\in \mathcal{F}(\Omega, X)$$\mathcal{B}(X)$$\xi,$$\xi$$\eta\in \mathcal{F}(\Omega, X)$

is measurable with respect to the -fields
Proposition I.1.9]. Thus, we say that a sequence
if
and write
measure to a separably valued. We also say thatfor every:$\mathcal{A}$$\omega\in\Omega\mapsto d(\xi(\omega), \eta(\omega))$$\sigma$andconverges in$\mathcal{B}(\mathbb{R})[23,$$\{\xi_{n}\}_{n\in N}\subset \mathcal{F}(\Omega, X)$$\xi_{n}arrow^{P}\xi$$\xi\in \mathcal{F}(\Omega, X)$$\Omega$$\epsilon>0$$d(\xi_{n}(\omega), \xi(\omega))\geq\epsilon\})=0$

distribution to$\xi$$\lim_{narrow\infty}P(\{\omega\in$$\xi_{n}$converges in

if $Po\xi_{n}^{-1}arrow^{L}Po\xi^{-1}.$

Theorem 8. Let$P:\mathcal{A}arrow[0, \infty)$$\{\xi_{n}\}_{n\in N}\subset \mathcal{F}(\Omega, X)$

is $co$ -continuous. If

andbe an autocontinuous nonadditive measure. Let. Assume that is separably valued and$\xi\in \mathcal{F}(\Omega, X)$$\xi_{n}arrow^{P}\xi$$P\circ\xi^{-1}$$\xi$, then $Po\xi_{n}^{-1}arrow^{L}Po\xi^{-1}.$

Remark 4. (1) By the same proof as for [15, Theorem 3.3], we can show that the
autocontinuity of is a necessary and sufficient condition for $Po\xi_{n}^{-1}arrow^{L}Po\xi^{-1}$ forsatisfying
and $Po\xi^{-1}$andany sequenceis co-continuous.
$P$$\{\xi_{n}\}_{n\in N}\subset \mathcal{F}(\Omega, \mathbb{R})$$\xi\in \mathcal{F}(\Omega, \mathbb{R})$$\xi_{n}arrow^{P}\xi$55(2) Letand. Denneberg [4, page 97] andMurofushi and co-workers [15] defined the notion of convergence in distribution
as $P(\xi_{n}>t)arrow P(\xi>t)$ except at most countably many values . It is easy toshow that the L\’evy convergenceand the co-continuity of
imply convergence in distribution in the sense of Denneberg. However, the converse
is not true.
$\{\xi_{n}\}_{n\in \mathbb{N}}\subset \mathcal{F}(\Omega, \mathbb{R})$$\xi\in \mathcal{F}(\Omega, \mathbb{R})$$t$$P\circ\xi_{n}^{-1}arrow^{L}Po\xi^{-1}$

Example 7. For each$n\in \mathbb{N}$$P\circ\xi^{-1}$, we define the nonadditive measure$\mu_{n}$:$\mathcal{B}(\mathbb{R})arrow[0,1]$

as$\mu_{n}(E):=$for every
L\’evy converge to$E\in \mathcal{B}(\mathbb{R})$

Theorem 9. Let
sure. Let$\{\begin{array}{ll}\delta_{-1}(E) if (t, \infty)\subset E for some t\in \mathbb{R}\delta_{-1}(E)/n if (t, \infty)\not\subset Efor any t\in \mathbb{R}\end{array}$. Then$\mu_{n}((t, \infty))arrow\mu((t, \infty))$for every$t\in \mathbb{R}$, but$\mu_{n}$does not$\mu.$

be a uniformly autocontinuous nonadditive mea. Assume that eachandis Radonandis $co$ -continuous. Furthermore, assume that
. The
following conditions are equivalent:$P:\mathcal{A}arrow[0, \infty)$$\{\xi_{n}\}_{n\in \mathbb{N}}\subset \mathcal{F}(\Omega, X)$$\xi\in \mathcal{F}(\Omega, X)$$P\circ\xi_{n}^{-1}$$P\circ\xi_{n}^{-1}arrow^{L}P\circ\xi^{-1}$$P\circ\xi^{-1}$(i) $Po\xi^{-1}$ is tight.

(ii)$\{P\circ\xi_{n}^{-1}\}_{n\in \mathbb{N}}$is uniformly tight.

7. CONCLUSION
We formalized the L\’evy-Prokhorov metric and the Fortet-Mourier metric for nonadditive measures on a metric space and showed that the L\’evy topology on every
uniformly equi-autocontinuous set of Radon nonadditive measures can be metrized
by such metrics. This result was proved using the uniformity for the L\’evy convergence on a bounded subset of Lipschitz functions. In applications to nonadditive
measure theory, we showed that convergence in measure implies convergence in distribution for a sequence of measurable mappings on an autocontinuous measurespace. We also showed the uniform tightness of a L\’evy convergent sequence of measurable mappings on a uniformly autocontinuous measure space. Our results could
supply weak convergence methods for related fields, such as Choquet expected utility
theory, game theory, and some economic topics under Knightian uncertainty.
An open problem is the fact that the L\’evy topology can be metrized by the L\’evyProkhorov metric and $/or$ the $Fortet-Mo$ urier metric on the set of all Radon autocontinuous nonadditive measures, which is not necessarily uniformly equi-autocoptinuous.

56
REFERENCES[1] A.D. Alexandroff, Additive set-functions in abstract spaces, Math. Sbomic 9 (51) (1941) 563628.
[2] P. Billingsley, Convergence of Probability Measures, second edition, John Wiley & Sons, New
York, 1999.
[3] G. Choquet, Theory of capacities, Ann. Inst. Fourier Grenoble 5 (1953-54) 131-295.
[4] D. Denneberg, Non-Additive Measure and Integral, second edition, Kluwer Academic Publishers, Dordrecht, 1997.
[5] R.M. Dudley, Real Analysis and Probability, Wadsworth & Brooks/Cole, California, 1989.
[6] R. Engelking, General Topology, Heldemann Verlag, Berlin, 1989.
[7] R. Fortet, E. Mourier, Convergence de la r\’epartition empirique vers la r\’epartition th\’eorique,Ann. Sci. Ecole Norm. Sup. 70 (1953) 267-285.
[8] I. Gilboa, D. Schmeidler, Updating ambiguous beliefs, J. Econ. Theory 59 (1993) 33-49.
[9] B. Girotto, S. Holzer, Weak convergence of masses on normal topological spaces, Sankhyaa(1993) 188-201.
[10] B. Girotto, S. Holzer, Weak convergence of bounded, monotone set functions in an abstract
setting, Real Analysis Exchange 26 (2000/2001) 157-176.
[11] J. Kawabe, The Alexandroff theorem for Riesz space-valued non-additive measures, Fuzzy
Sets and Systems 158 (2007) 2413-2421.
[12] J. Kawabe, Regularities of Riesz space-valued non-additive measures with applications toconvergence theorems for Choquet integrals, Fuzzy Sets and Systems 161 (2010) 642-650.
[13] J. Kawabe, Metrizability of the L\’evy topoloy on the space of nonadditive measures on metricspaces, Fuzzy Sets and Systems (2012), doi:10. $1016/j.fss$ .2012.02.005.
[14] R. Kozhan, Non-additive anonymous games, Int. J. Game Theory 40 (2011) 215-230.
[15] T. Murofushi, M. Sugeno, M. Suzaki, Autocontinuity, convergence in measure, and convergence in distribution, FUzzy Sets and Systems 92 (1997) 197-203.
[16] T. Norberg, Random capacities and their distributions, Prob. Theory Relat. Fields 73 (1986)

281-297.

[17] Y. Namkawa, T. Murofushi, M. Sugeno, Space of fuzzy measures and convergence, Fuzzy Sets
and Systems 138 (2003) 497-506.
[18] Y. Namkawa, T. Murofushi, Choquet integral with respect to a regular non-additive measure,in: Proceedings of the 2004 IEEE Intemational Conference on Fuzzy Systems (FUZZ-IEEE
2004), 2004, pp. 517-521.
[19] K.G. Nishimura, H. Ozaki, Search and Knightian uncertainty, J. Econ. Theory 119 (2004)

299-333.

[20] G.L. ’Brien, Sequences of capacities, with connections to large-deviation theory, J. Theor.
Prob. 9 (1996) 19-35.
[21] K.R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York, 1967.
[22] E. Pap, Null-Additive Set Functions, Kluwer Academic Publishers, Dordrecht, 1995.
[23] N.N. Vakhania, V.I. Tarieladze, S.A. Chobanyan, Probability Distributions on Banach Spaces,
D. Reidel Publishing Company, 1987.
[24] V.S. Varadarajan, Measures on topological spaces, Am. Math. Soc. Transl. Ser. 2, 48 (1965)$O$

161-228.

[25] Z. Wang, G.J. Khr, Generalized Measure Theory, Springer, New York, 2009.
[26] G. Wang, X. Li, On the weak convergence of sequences of fuzzy measures and metric of fuzzymeasures, Fuzzy Sets and Systems 112 (2000) 217-222.

DEPARTMENT OF MATHEMATICSFACULTY OF ENGINEERINGSHINSHU UNIVERSITY
4-17-1 WAKASATO, NAGANO 3SO-S553, JAPAN
$E$-mail address: jkawabe@shinshu-u.ac.jp