数理解析研究所講究録 1452 巻 2005 年 1-10

1Relationship among continuity conditions
and null-additivity conditions
in non-additive measure theory *朝比奈 伸 , 内野 健太, 室伏 俊明
Asahina,
Uchino Kenta, Toshiaki Murofushi
Shin

東京工業大学大学院総合理工学研究科知能システム科学専攻
Department of Computational Intelligence and Systems Science,
Tokyo Institute of Technology

Abstract
The paper shows all the implication relationship among six continuity conditions and two
null-additivity conditions with respect to non-additive measures. The six continuity conditions
are continuity from above, order continuity, strong order continuity, exhaustivity, continuity, and null-continuity, and the two null-additivity conditions are null-additivity andfromnull-additivity.
Furthermore the implication relationship is summarized in a diagram.
weak$\mathrm{b}\mathrm{e}_{1}^{\rceil}\mathrm{o}\mathrm{w}$1Introduction

So far, non-additive measure theory has been constructed along lines of the classical measure
theory $[1, 12]$ ; here a non-additive measure means a non-negative monotone set function. In general,
theorems in the classical measure theory do not hold in non-additive measure theory without
measures haveadditional conditions. Therefore several conditions with respect tobeen given in order that such theorems hold [2, 3, 6, 7, 8, 11, 12]. For example, Egoroff’s theorem,
which is one of the most important convergence theorems in the classical measure theory, does nothold in non-additive measure theory, but it holds under continuity from above and below, that is,the continuity from above and below is a sufficient condition for the theorem [7]. In the same way,
sufficient conditions for other theorems have been given [2, 3, 6, 7, 8, 11, 12], but the relationshipamong these conditions have hardly been studied so far. For this reason, it is very important toclarify the relationship among them.
In this paPer, we investigate eight well-known conditions with respect to continuity and nutladditivity, and show all the implication relationship among them.
This paper is organized as follows: In Section 2, we give the definitions of a non-additive measure
and the eight conditions discussed in this paper. In Section 3, the main results are stated. We give
all the implication relations among the eight conditions, and show through ten counterexamples
that there exist no further implication relations. In Section 4, we state the conclusion of this paper,and give a subject of future research.
$\mathrm{n}\mathrm{o}\mathrm{n}rightarrow \mathrm{a}\mathrm{d}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}$“This work is partially supported by a grant from the Ministry of Education, Culture, Sports, Science andTechnology, the 21st Century COE Program “Creation of Agent-Based Social Systems Sciences.

2
Notation and definitions

2

Let $X$ be a non-empty set, be a a-algebra of subsets of $X$ , and let denote the set of all positive
integers.
According to Murofushi [8], we define a non-additive measure as follows:$\mathrm{N}$$\mathrm{f}\mathrm{f}$Definition 1 A non-additive measure on
two conditions:(i)$\mu(\emptyset)=0$

(ii)

$A$

,

$\mathfrak{F}$is a set function

$\mu$

:

$\mathrm{f}\mathrm{f}$satisfying the following$arrow[0, \infty]$,$B\in \mathrm{f}\mathrm{f}$, $A\subset B\Rightarrow\mu(A)\leq\mu(B)$ .

A non-additive measure$\mu$

is said to be finite if $\mu(X)<\infty$ .

Unless stated otherwise, all subsets are supposed to belong to ff andnon-additive measure on if$\mu$

is assumed to be a$\cdot$

The following conditions concerning null-additivity are defined by Wang and Klir.

Definition 2(i)$\mu$is said to be null-addition, if $\mu(A\cup B)=\mu(A)$ for every set $B$ such that$\mu(B)=0[11]$ .

(ii)

$\mu$

is said to be weakly null-additive if $\mu(A\cup B)=0$ for every sets$A$

and $B$ such that$\mu(A)=\mu(B)=0$ $[12]$ .

The following conditions concerning continuity are defined by Drewnowski [4], Jiang et al. [5]and, Uchino and Murofushi [9].

Definition 3(i)$\mu$is said to be continuous from above [resp. below] if$\mu(\lim_{narrow\infty}A_{n})=\lim_{narrow\infty}\mu(A_{n})$for every decreasing [resp. increasing] sequence
if it is continuous both from above and below.
(ii). In addition,(v)

is said to be continuous$\mathrm{h}.\mathrm{m}_{narrow\infty}\mu(A_{n})=0$$A_{n}\downarrow\emptyset$

is said to be strongly order continuous if it is continuous at measurable sets of measure zero,.
that is, $\lim_{narrow\infty}\mu(A_{n})=0$ for every decreasing sequencesuch that$\mu$$\{A_{n}\}$

(iv)

$\mu$

is said to be order continuous if it is continuous at the empty set, that is,[4].
for every decreasing sequencesuch that$\mu$$\{A_{n}\}$(iii)$\{A_{n}\}$is said to be null-continuous ifthat $\mu(A_{n})=0$ for every n[9].
$\mu$

$\mu$

$\mu(\bigcup_{n=1}^{\infty}A_{n})=0$

for$\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{y}\backslash$$\mu(\bigcap_{n=1}^{\infty}A_{n})=0[5]$increasing sequence$\{A_{n}\}$is said to be exhaustive if $\lim_{narrow\infty}\mu(A_{n})=0$ for every disjoint sequence $\{A_{n}\}[4]$ .

Note that every classical finite measure satisfies all the conditions stated above

such

3

OI-add : null-additivity: continuity from above: order continuity: continuity from below$\backslash \mathrm{v}- \mathrm{O}- \mathrm{a}\{\mathrm{l}\mathrm{d}$$\downarrow 0$$\downarrow$: weak null-additivity: strong order continuity

exh : exhaustivity$\downarrow\emptyset$

10 : null-continuity$\uparrow$

Figure 1: Implication relationship3

All the implication relationship among the eight conditions

In this section, we show all the implication relationship among the eight conditions stated in
Definitions 2 and 3. We first represent the relations among them in a diagram as Fig. 1, then we
discuss the implication relations, and show through ten counterexamples that there exist no other
implication relations.

means that
Figure 1 summarizes all the relations. In this diagram, an arrow from A to”.
indicates the proposition showing it; forcondition A implies condition , and its label
example, the fact that continuity from above implies strong order continuity is shown by Proposition 1 (ii). The arrow corresponding to a trivial relation has no label. On the other hand, theabsence of a directed path means that A does not imply ; for example, null-continuity does not
imply continuity from below, and continuity from below does not imply null-additivity$\mathrm{B}$$\mathrm{B}$$” \mathrm{P}\mathrm{r}$$\mathrm{X}$$\mathrm{B}$

4

3.1

Implication Relationship$|$We first state all the implication relations rom one condition to another; most of these implication
relations are obvious.
Proposition 1 (i) If is null-additive, then it is weakly null-additive.
$\mathrm{f}$$\mu$(ii) a) If
b) if
c) if
(iii) If

$\mu$

$\mu$

$\mu$

$\mu$

is continuous from above, then it is strongly order continuous,is strongly order continuous, then it is order continuous,
is order continuous, then it is exhaustive [4].

is continuous from belouz, then it is reull-continuous.

Then we state the implication relations from more than one condition to another.

Proposition 2 If
Proposition 3 If

$\mu$

$\mu$

is exhaustive and continuous from below, then it is order continuous [3].
is null-additive and order continuous, then it is strongly order continuous [6].

By Propositions 2 and 3, we have the following corollary.
Corollary 1 If
continuous.

is null-additive, exhaustive and continuous from below, then it is strongly order$\mu$

Proposition 4 If$\mu$

is null-additive and 0 rder continuous, then it is null-continuous. [10]The following proposition is our new result.

Proposition 5 If is weakly null-additive and strongly order continuous, then it is null-continuous.
$\mu$

and
Proof. Assume that is weakly null-additive and strongly order continuous. Let
recursively as follows. Letof$\mu(N_{n})=0$ for all $n\geq 1$ . We first define a subsequence
as $narrow\infty$ , by strong order$n_{1}=1$ . For $m\geq 1$ , since $\mu(N_{n_{m}})=0$ andcontinuity we can choose $n(m+1)$ so that $n(m+1)>n_{m}$ and$N_{n}\uparrow N$$\mu$$\{N_{n}\}$$\{N_{n_{m}}\}$$N_{n_{m}}\cup(N\backslash N_{n})\downarrow N_{n_{m}}$$\mu(N_{n_{m}}\cup(N\backslash N_{n_{(m+1)}}))$$< \frac{1}{m}$.

Then we define$A= \bigcup_{1}(N_{n_{2i}}\backslash N_{n_{(2i-1)}})):=\infty$

For every , since$\mathrm{i}$$A\subset N_{n_{2i}}\cup$(A$\backslash N_{n_{(2i+1)}}$$B=N\backslash A$.

), it follows that$\mu(A)\leq\mu(N_{n_{2i}}\cup$( $N$ A

$N_{n_{(2\mathrm{i}+1)}}$)$)$$< \frac{1}{2\dot{\iota}}$.

Hence $\mu(A)=0$ . Similarly we have $\mu(B)=0$ . Then weak null-additivity implies that$\mu(N)=\mu(A\cup B)=0$ .

Therefore pa is null-continuous.

$\square$

There exist no implication relations except stated above. We show it through ten counterexampies given in the following subsection

5$\frac{\mathrm{T}\mathrm{a}\mathrm{b}1\mathrm{e}1^{\sim}\mathrm{S}\mathrm{u}\mathrm{m}1\mathrm{n}\mathrm{a}\mathrm{r}\mathrm{y}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{c}\mathrm{x}\mathrm{a}\mathrm{n}1\mathrm{p}1\mathrm{e}\mathrm{s}}{\ovalbox{\tt\small REJECT}}\mathrm{E}\mathrm{x}\mathrm{a}\mathrm{r}\mathrm{n}\mathrm{p}1\mathrm{c}0- \mathrm{a}\mathrm{d}\mathrm{d}.\mathrm{v}\mathrm{v}- \mathrm{O}- \mathrm{a}\mathrm{d}\mathrm{d}\downarrow\downarrow 0^{\cdot}\downarrow\emptyset \mathrm{e}\mathrm{x}$$\}_{1}$

Ex. 1
EX. 2
Ex. 3
Ex. 4/
Ex. 5
Ex. 6
Ex. 7
Ex. 8
Ex. 9
Ex. 10$\mathrm{F}$F$\mathrm{F}$F$\mathrm{T}$T$\mathrm{T}$T$\mathrm{T}$T$\mathrm{T}$$\uparrow 0$T$\mathrm{T}$$\mathrm{T}$$\mathrm{F}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$\mathrm{F}$$\mathrm{F}$$\mathrm{F}$$\mathrm{F}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$\mathrm{F}$$\mathrm{F}$$\mathrm{F}$$\mathrm{T}$$\mathrm{F}$$\mathrm{T}$$\mathrm{F}$$\mathrm{T}$$\mathrm{F}$$\mathrm{F}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$\mathrm{k}^{\urcorner}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$\prime 1^{\urcorner}$$\mathrm{F}$$\mathrm{F}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$\mathrm{F}$$\mathrm{F}$$\mathrm{F}$$\mathrm{F}$$\mathrm{T}$$\mathrm{F}$$\mathrm{F}$$\mathrm{T}$$\mathrm{T}$$\mathrm{F}$$\mathrm{T}$$\mathrm{T}$$\mathrm{F}$$\mathrm{F}$$\mathrm{F}$$\mathrm{T}$$\mathrm{F}$$\mathrm{F}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$\mathrm{F}$$\mathrm{T}$O-add :null-additivity: continuity from above
4 : order continuity: continuity from below$\downarrow$$\emptyset$$\uparrow$

3.2

T$\uparrow$w-O-add : weak null-additivity: strong order continuity
exh : exhaustivity: null-continuity$\downarrow 0$$\uparrow 0$

Examples with respect to non-additive measures

The absence of other arrows in Fig. 1 is shown by counterexamples, Examples 1-10, in this subsection. Table 1 summarizes all the examples. In this table, the symbol “ ” in the cell at the row ofand the column of a condition “$C’$ means that the non-additive measure in Example A“Ex.
satisfies the condition , and the symbol ” means that it does not; for instance, the non-additive
measure in Example 6 satisfies all conditions except continuity from above.
$\mathrm{T}$)”$\mathrm{A}$$C$$” \mathrm{F}$In the following two examples, non-additive measures are defined on a finite set. Obviously
such non-additive measures satisfy all the continuity conditions.
Example 1 Let $X=\{0,1\}_{1}$ and$\mu$

be the non-additive measure on the power set

as, {0} or {1}0if
1if $A=X$ .

$2^{X}$

of $X$ defined$A=\emptyset$$\mu(A)=\{$$)$

is not weakly null-additive; indeed, if we take $A=\{0\}$ and $B=\{1\}_{7}$ then $\mu(A)=$$\mu(B)=0$ , but we have $\mu(A\cup B)=\mu(X)=1$ by the definition of . It follows from this that it is
not null-additive either. On the other hand, obviously satisfies the other conditions.
Then

$\mu$

$\mu$

$\mu$

Example 2 Let $X=\{0, 1\}$ , and$\mu$

be the non-additive measure on the power set

as$2^{X}$

of $X$ definedor {0} ,if
if $A=\{1\}$ ,if $A=X$ .
$\mathrm{A}=\emptyset$$\mu(A)=\{$

1
2

Then is not null-additive; indeed, if we take and $B$ as in Example 1, then $\mu(A)=0$ , but
we have $\mu(A\cup B)=\mu(X)=2\neq 1=\mu(B)$ by the definition of . On the other hand, obviously
satisfies the other conditions.
$A$

$\mu$

$\mu$

In the remaining examples, non-additive measures are defined on a countable set$\mu$$\mathrm{G}$Example 3 Let$X=\mathrm{N}$

, and

$\mu$

be the non-additive measure on the power set$\mu(A)=\{$

if
0
1if

$A=\emptyset$$A\neq\emptyset$$2^{X}$

of $X$ defined as,.

is a disjoint
Then pa is not exhaustive. Indeed if we take $A_{n}=\{n\}$ for $n=1,2$ , , then$\lim_{narrow\infty}\mu(A_{n})=1$.
sequence and $\mu(A_{n})=1$ for every by the definition of . Hence we haveThus is not exhaustive. It follows from this that it does not satisfy the following three conditions:continuity from above, strong order continuity and order continuity. On the other hand, obviously
satisfies the other conditions.
$\ldots$$n$$\{A_{n}\}$

$\mu$

$\mu$

$\mu$

Example 4 Let
of $X$ defined as$X=\mathrm{N}$, $m(A)$$= \sum_{n\in \mathrm{A}}1/2^{n}$, andbe the non-additive measure on the power set$\mu$$2^{X}$if
if$m(A)$$\mu(A)=\{$

1$|A^{\mathrm{c}}|=\infty$,$|A^{\mathrm{c}}|<\infty$.

is not order continuous; indeed, if we take $A_{n}=\{n, n+1, , . .\}$ for $n=1,2$ , , thenfor any , but we have $\lim_{narrow\infty}\mu(A_{n})=1$ by the definition of .
A as $narrow\infty$ and
It follows from this that it is neither continuous from above nor strongly order continuous. On the
be a disjoint sequence. If $\mu(A_{n})<1$ forother hand, we can show that is exhaustive. Let$\lim_{narrow\infty}\mu(A_{n})=0$ since $m$ is a finite.
Thus$\mu(A_{n})=m(A_{n})$definition
of
by
the
every
then
by
measure. If not, that is, there exits a positive integer such that $\mu(A_{l})=1$ , then$n\neql$.
Hence
for any such thatis disjoint, we havethe definition of , and since$\mathrm{m}(\mathrm{A})=m(A)$ by
oo for any $n\neq l$ . Then, similarly we have $\lim_{narrow\infty}\mu(A_{n})=0$ . Therefore
is exhaustive. By the fact that is not order continuous but exhaustive, and Proposition 2, it
is not continuous from below. Since $m$ is a finite measure, obviously is null-continuous and nulladditive, hence it is also weakly null-additive.
Then,$\ldots$

$\mu$

$n$

$|A_{n^{\mathrm{C}}}|<\infty$$A_{n}\downarrow$$\mu$$\{A_{n}\}$

$\mu$

$\mu$

$n_{1}$$|A_{l^{\mathrm{C}}}|<\infty$$l$$A_{n}\subset A\iota^{c}$$\{A_{n}\}$

$\mu$

$n$

$|A_{n^{\mathrm{C}}}|=$

$\mu$

$\mu$

$\mu$

Example 5 Let
of $X$ defined as$X=\mathrm{N}$, $m(A)= \sum_{n\in A}1/2^{n}$ , and$\mu$

be the non-additive measure on the power set$2^{X}$$\mu(A)=\{$

0if$m(A)$
if$\mathrm{A}=\{1\}$,$A\neq\{1\}$.

is not strongly order continuous; indeed, if we take $A_{n}=\{1\}\cup\{n+1, n+2, \ldots\}$ for, but we have $\lim_{narrow\infty}\mu(A_{n})=1/2$
as, and $A=\{1\}$ , then$n=1,2$ ,andby the definition of . It follows from this that it is not continuous from above either. On the otheris clearly
hand, obviously is weakly null-additive. Furthermore since $m$ is a finite measure,order continuous and continuous from below, thus it is also exhaustive and null-continuous. By
the fact that is not strongly order continuous but order continuous, and Proposition 3, it is notnull-additive.
Then

$\mu$

$A_{n}\downarrow A$$\ldots$$narrow\infty$$\mathrm{m}(\mathrm{A})=0$

$\mu$

$\mu$

$\mu$

$\mu$

Example 6 Let
of $X$ defined as$X=\mathrm{N}$, $m(A)$ $= \sum_{n\in A}1/2^{n}$ , and$\mu$

be the non-additive measure on the power set$2^{X}$

if $m(A)\leq 1/2$ ,1if $m(A)>1/2$ .

$m(A)$$\mu(A)$$=\{$and as in Example 5, then
Then is not continuous from above; indeed, if we takeoo and $\mu(A)=1/2$ , but we have $\mu(A_{n})=1$ for every by the definition of . On the other
hand, since $m$ is a finite measure, obviously satisfies thhe other conditions.
$A_{n}$$\mu$as$narrow$

$n$

$\mu$

$A$

$A_{n}\downarrow A$

$\mu$

7

Example 7 Let$X=\mathrm{N}$

, and

$\mu$

be the non-additive measure on the power set$\mu(A)=\{$$2^{X}$

of $X$ defined as

0if $A\neq X$ ,1if $A=X$ .

$X$
Then is not null-continuous; indeed, if we take $A_{n}=\{1,2, . ., , n\}$ for $n=1$ , 2, , thenand $\mu(A_{n})=0$ for every , but we have $\mu(X)=1$ by the definition of . It follows from this that
it is not continuous from below either. On the other hand, obviously is continuous from above.
It follows from this that it satisfies strong order continuity, order continuity and exhaustivity. By
the fact that is not null-continuous but strongly order continuous, and Proposition 5, it is not
weakly null-additive. Hence it is not null-additive either.
$\ldots$

$\mu$

$n$

$A_{n}$$\uparrow$

$\mu$

$\mu$

$\mu$

Example 8 Let$2^{X}$$X=\mathrm{N}$, $m(A)$ $= \sum_{n\in A}1,/2^{n}$ , andof $X$ defined as$\mu(A)=\{$$\mu$

be the non-additive measure on the power set

0if$|A|<\mathrm{m}$,if$|A|=\infty$.

$m(A)$

Then, similarly in Example 7, is not null-continuous. Hence it is not continuous from below
either. On the other hand, obviously is weak null-additive and order continuous. Hence it has
exhaustivity. By the fact that is not null-continuous but order continuous, and Proposition 4,
it is not null-additive. Furthermore, similarly in Example 5, is not strongly order continuous.
Hence it is not continuous from above either.
$\mu$

$\mu$

$\mu$

$\mu$

,
be a non-principal ultrafilter on $X$ , i.e., an ultrafilter containing the
Example 9 Let
of $X$
cofinite filter $\{M\subset X||M^{c}|<\infty\}$ , and be the non-additive measure on the power set
defined as,
0if$\mu(A)=\{$.
1if$X=\mathrm{N}$$\mathfrak{U}$$2^{X}$$\mu$$A\not\in \mathfrak{U}$$A\in \mathfrak{U}$Then, similarly in Example 7, is not null-continuous. Hence it is not continuous from below
either. On the other hand, we can show that is null-additive. Let be a set such that $\mu(A)=0$ .
since 2[ is an ultrafilter.
by the definition of . If $B$ ( , then we haveThen
, then we have $A\cup B$ (? . Therefore we haveThus we have $\mu(A\cup B)=0=\mu(B)$ . If
is null-additive. It follows from this that it is also weakly null$\mu(A\cup B)=1=\mu(B)$ . Hence
additive. By the fact that is not null-continuous but nulJ-additive, and Proposition 4, it is notorder continuous. Therefore it is neither strongly order continuous nor continuous from above.
be a disjoint sequence. If $\mu(A_{n})=0$
Furthermore we can show that is exhaustive. Let$\lim_{narrow\infty}\mu(A_{n})=0$
is,
there
exists positive integer $m$ such that
. If not, thatfor any , thenis disjoint and 2[ is a filter, it
by the definition of . Since$\mu(A_{m})=1$ , then we have
( for every $n\neq m$ , and hence that $\mu(A_{n})=0$ for every $n\neq m$ . Thus we havefollows that
iim $narrow\infty\mu(A_{n})=0$ . Therefore pa is exhaustive.
$\mu$

$A$

$\mu$

$A\cup B\not\in \mathfrak{U}$$\mathfrak{U}$$A\not\in \mathfrak{U}$$\mu$$B\in \mathfrak{U}$$\mathfrak{U}$

$\mu$

$\mu$

$\{A_{n}\}$

$\mu$

$n$

$A_{m}\in \mathfrak{U}$$A_{n}$$\mu$$\{A_{n}\}$$\mathfrak{U}$Example 10 Let
of $X$ defined as
set$X=\mathrm{N}$, $m(A)$ $= \sum_{n\in A}1/2^{n}$ , and$\mu$

be the non-additive measure on the power$2^{X}$$\mu(A)=\{$$m(A)$

2

if $m(A)<1$ ,if $m(A)=1$ .

8Table 2: Boolean matrix of implication relationship among eight conditions$\ovalbox{\tt\small REJECT}_{1\downarrow 0\downarrow\emptyset \mathrm{e}\mathrm{x}\mathrm{h}\uparrow}0_{-\mathrm{a}\mathrm{d}\mathrm{d}\mathrm{w}- 0- \mathrm{a}\mathrm{d}\mathrm{d}}\uparrow 0$–$\mathrm{O}\mathrm{a}\mathrm{d}\mathrm{d},\downarrow_{7})$$0- \mathrm{a}\mathrm{d}\mathrm{d},\downarrow,)$

O-add,
0-add,J,l0–$\downarrow,\uparrow 0$

O-add,lO-add,l$0,\uparrow$–$0,\uparrow 0$–$\mathrm{O}- \mathrm{a}\mathrm{d}\mathrm{d}_{\backslash }\downarrow \mathrm{O},\uparrow$$\mathrm{O}-\mathrm{a}\mathrm{d}\mathrm{d},\downarrow \mathrm{O},\uparrow \mathrm{O}$O-add,, 10-.aO-add,l0-dr$\mathrm{d}\mathrm{d}$—-$\emptyset,\uparrow$–$\mathrm{d}\mathrm{d},\downarrow\emptyset,\uparrow0$$\emptyset,\uparrow 0$$\mathrm{O}- \mathrm{a}\mathrm{d}\mathrm{d},\mathrm{e}\mathrm{x}\mathrm{h},\uparrow$0\mathrm{a}\mathrm{d}\mathrm{d},\mathrm{e}\mathrm{x}\mathrm{h},\uparrow$\mathrm{O}$$\mathrm{O}- \mathrm{a}\mathrm{d}\mathrm{d},\mathrm{e}\mathrm{x}\mathrm{h},\uparrow$\mathrm{w}\mathrm{d}\mathrm{d},\downarrow,\uparrow$$\mathrm{w}-\mathrm{O}-_{\acute{\mathrm{e}}}\iota\mathrm{O}-_{\acute{C}}\iota \mathrm{d}\mathrm{d},\downarrow,\uparrow$w-O-add,
w-O-add,l)w-Ow-O-.a
w-O-add,X$1,\mathrm{t}0$$\mathrm{F}$$\mathrm{w}-\mathrm{O}- \mathrm{a}\mathrm{d}\mathrm{d},\downarrow \mathrm{O},\uparrow$$0,\uparrow$$\mathrm{b}^{\neg}$$\mathrm{F}$$\mathrm{w}- \mathrm{O}- \mathrm{a}\mathrm{d}\mathrm{d},\downarrow,\uparrow \mathrm{O}$0$$,\uparrow 0$$.\mathrm{a}\mathrm{d}\mathrm{d},\downarrow[$$\mathrm{d}\mathrm{d},\downarrow[),\uparrow$\mathrm{w}- \mathrm{C}-_{\mathrm{c}}\backslash \mathrm{d}\mathrm{d},\rfloor$$\emptyset,\uparrow$$\phi,\uparrow$w-O-add,lw-O-add,exh,1$\mathrm{w}- \mathrm{O}- \mathrm{a}\mathrm{d}\mathrm{d},\downarrow\emptyset?\uparrow
0$ 0$$\emptyset_{?}\uparrow$\backslash \mathrm{v}- \mathrm{O}- \mathrm{a}\mathrm{d}\mathrm{d},\mathrm{e}\mathrm{x}\mathrm{h},]$w-O-a{;d.exh,\uparrow 0$\mathrm{F}$$\mathrm{F}$$b^{\backslash }$$\mathrm{F}$$\mathrm{F}$–$\mathrm{T}$$(1^{\supset}\mathrm{r}.1)$$\mathrm{T}(1^{\supset}\mathrm{r}.1)$$\mathrm{T}$–$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$(\mathrm{P}\mathrm{r}.1)$$\mathrm{T}(\mathrm{P}\mathrm{r}.1)$$\mathrm{T}$(Pr.l)$\mathrm{F}$$(\mathrm{P}\mathrm{r}.1)$$\mathrm{b}^{\urcorner}$$(\mathrm{P}\mathrm{r}.1)$$\mathrm{T}(\mathrm{P}\mathrm{r}.1)$$\mathrm{F}$$\mathrm{T}$–$\mathrm{T}$–$\mathrm{T}$(Pr.l)$(\mathrm{P}\mathrm{r} 1)$)–T(Ex.2)(Ex.2)(Ex.2)(Ex.2)(Ex.2)(Ex.2)$—$(Ex.2)(Ex.2)–$\mathrm{F}$(Pr.l)$(\mathrm{P}\mathrm{r}.1)$$\mathrm{T}(\mathrm{P}\mathrm{r}.1)$$\mathrm{T}$$\mathrm{T}$$\langle \mathrm{P}_{1^{\backslash}}.1$$\langle\mathrm{P}\mathrm{r}.1)$$\mathrm{F}$$\mathrm{b}^{\urcorner}$(Ex 6)(Ex.6)$(\mathrm{F}_{\lrcorner}\mathrm{x}.6)$(Ex.6)(Ex.6)(Ex.6)$(\mathrm{F}_{\lrcorner}’\mathrm{x}.6)$

F (Ex 6)$\mathrm{F}$$(\mathrm{b}^{\urcorner}\mathrm{x} 6)$–—–$(\mathrm{F}\lrcorner \mathrm{x}.2)$—––$(\mathrm{b}^{\urcorner}\mathrm{x}.2)$–$\mathrm{F}$–$\mathrm{F}$–$\mathrm{F}$–$\mathrm{F}$–$\mathrm{F}$$(\mathrm{E}\mathrm{x}.2)$–$\mathrm{F}$(Ex.6)(Ex.6)(Bx.6)(Ex.6)(Ex.6)(Ex.6)(Ex.6)

O-add :null-additivity: continuity from above: order continuity: continuity from below$\downarrow$$\downarrow\emptyset$$\uparrow$$\mathrm{T}$–$\mathrm{T}$$\mathrm{T}$–$\mathrm{T}$(Pr.3)(Pr.3)(Cor. 1)
F (Ex.4)(Pr.l)(Pr.l)$\mathrm{F}$$(\mathrm{E}^{\tau}\mathrm{x}.4)$$\mathrm{T}$$\mathrm{F}$$\mathrm{F}$$\mathrm{F}$$\mathrm{F}$—-(Pr.l)–$\mathrm{T}$$(\mathrm{C}\mathrm{o}I^{\cdot}.1)$$\mathrm{T}$$(\mathrm{P}_{1}.1)$–$\mathrm{T}$$(\mathrm{b}^{\neg}\mathrm{x}.6)$$(\mathrm{b}^{\urcorner}\mathrm{x}.6)$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$(Pr.l)$(\mathrm{P}\mathrm{r}.?)$$)$$(\mathrm{P}\mathrm{r}. 1)$$\mathrm{T}(\mathrm{P}\mathrm{r}.1)$$(\mathrm{I}^{3}1^{\backslash}.1)$$(\mathrm{P}_{1^{\backslash}}.1)$$\mathrm{E}^{\backslash }$$\mathrm{F}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$(\mathrm{P}\mathrm{r}.])$$\mathrm{T}(\mathrm{P}\mathrm{r}.1)$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$$\mathrm{T}$(Pr.l)––$(\mathrm{b}^{\urcorner}\mathrm{x}.5)$–$\mathrm{F}$–$\mathrm{T}$$(\mathrm{P}\mathrm{r}.1)$$\mathrm{T}(\mathrm{P}\mathrm{r}.1)$–$(\mathrm{E}\mathrm{x}^{r}.10)$–$\mathrm{T}$–—–$\mathrm{T}$$(\mathrm{P}\mathrm{r}.1)$–$\mathrm{T}$$(\mathrm{E}\prime \mathrm{x} 5)$$\mathrm{T}$$(\mathrm{P}\mathrm{r} 1)$$\mathrm{T}(\mathrm{F}\mathrm{r}1)$$\mathrm{F}$$\mathrm{F}$$\mathrm{F}$–$(\mathrm{P}\mathrm{r}.1)$$\mathrm{T}(\mathrm{P}\mathrm{r}.1)$$(\mathrm{P}\mathrm{r} 1)$$\mathrm{T}(\mathrm{P}\mathrm{r}1)$(Pr.l)$(\mathrm{P}\mathrm{r}.1)$(Ex.10)–$(\mathrm{P}_{1^{\backslash }}.1)$$\mathrm{T}(\mathrm{P}\mathrm{r}.1)$$\mathrm{F}$–$\mathrm{F}$$\mathrm{T}$(Ex.10)(Ex.lO)$(\mathrm{P}\mathrm{r}. 1)$$\mathrm{T}(\mathrm{P}\mathrm{r}.1)$$(\mathrm{P}\mathrm{r}.1)$–$\mathrm{T}$$(\mathrm{P}\mathrm{r}.1)$$\mathrm{T}(\mathrm{P}\mathrm{r}.1)$–$\mathrm{F}$$\mathrm{T}$$\mathrm{T}$(Pr.l)$(\mathrm{P}\mathrm{r}.1)$$\Gamma^{4}(\mathrm{F}\lrcorner \mathrm{x}.10)$$(\mathrm{P}\mathrm{r} 1)$$\mathrm{T}(\mathrm{P}\mathrm{r}1)$$\mathrm{T}$(Pr.l)–(Ex.10)–F (Ex.lO)(Ex.4)–(Ex 5)(Ex.5)(Ex.5)(Ex.5))(EX.

$\mathrm{T}$$\mathrm{T}$$(\mathrm{P}\mathrm{r} 1)$$\mathrm{T}(\mathrm{P}\mathrm{r}1)$$(\mathrm{P}\mathrm{r}.1)$$\mathrm{T}(\mathrm{P}\mathrm{r}.1)$(Ex, 10)(Ex.
(Pr.l)(Pr.l)–(Ex.10)(Pr.l)(Pr.l)–(Pr.l)(Ex.10)(Ex.lO)(Pr.l)$(\mathrm{P}\mathrm{r}.1)$$\mathrm{T}(\mathrm{P}\mathrm{r}.1)$(Pr.2)––$(\mathrm{P}\mathrm{r} 1)$$\mathrm{T}(\mathrm{P}\mathrm{r}1)$–$\mathrm{T}$$(\mathrm{P}\mathrm{r}.1)$$\mathrm{T}(\mathrm{P}\mathrm{r}.1)$–$(\mathrm{E}^{\neg}\mathrm{x}.10)$$\mathrm{F}(\mathrm{E}^{\neg}\mathrm{x}10)-\cdot$$\mathrm{T}(\mathrm{f}’ \mathrm{r}.1)--$$\mathrm{T}$$(\mathrm{P}\mathrm{r}.?.\cdot)$$\mathrm{T}(\mathrm{P}\mathrm{r}.1\cdot)$$(\mathrm{E}\mathrm{x}^{\Gamma}.5)$––$\mathrm{T}$$(\mathrm{P}\mathrm{r}.1)$$(\mathrm{P}\mathrm{r}.1)-$$r$$\mathrm{F}^{2}$$\mathrm{F}$$1\mathrm{J}$(Ex.4)–$\mathrm{F}$(Ex.10)$(\mathrm{F}_{\mathrm{J}}\mathrm{x}^{r}.10)$

w-O-add : weak null-additivity: strong order continuity
exh : exhaustivity: null-continuit$\downarrow 0$$\uparrow 0$–

9

as in Example 7 then
Then is not continuous from below. Indeed, if we takeand $m(A_{n})=1-1/2^{n}<1$ for any . However we have $\lim_{narrow\infty}\mu(A_{n})=1\neq 2=\mu(X)$ by
the definition of . Thus is not continuous from below. On the other hand, since $m$ is a finite
measure, obviously satisfies the other conditions.
$A_{n}\uparrow X$$A_{n}$

$\mu$

$n$

$\mu$

$\mu$

$\mu$

”
in
Table 2 illustrates the implication relationship among the eight conditions. The symbol“ $B$ ”“”
means that the condition
and the column of a condition
the cell at the row of a condition“”
implies the condition “ $B”$ . When there exist more than one condition in the row, they stand”indicates the proposition showing the implication;
for the conjunction of the conditions. “
for instance, the fact that exhaustivity and continuity from below together imply order continuity”
means that the condition ” does not imply theis shown by Proposition 2. The symbol“ $B”$)” indicates its counterexample. The implication relationship, and then “(Ex.
condition“–,,vacuously
true. Table 2 is a minimal expression of implicationisindicated by the symbol
relationship in the same that many other implication relations can be easily derived as a simple
application of Proposition 1. For example, the sufficient conditions implied by both null-additivity
and weak null-additivity are exactly those implied only by null-additivity.
$‘\zeta \mathrm{T}$

$A$

$A$

$\mathrm{n}\mathrm{d}$$(\mathrm{P}\mathrm{r}.

\mathrm{X})$$” A$$” \mathrm{F}$$\mathrm{X}$4

Concluding Remarks

We have showed all the implication relationship am ong the eight conditions with respect to nonadditive measures: continuity from above, order continuity, strong order continuity, exhaustivity,
continuity from below, null-continuity, null-additivity and weak null-additivity. Furthermore wehave summarized these implication relationship as in Fig. 1.
In this paper, we have dealt only with the eight conditions, but there exist many other conditions
which are given as a suffcient condition in order that a theorem holds. For this reason, we areresearching the implication relationship among other conditions such as auto-continuity, property, and pseudometric generating property.
$\mathrm{S}$References[1] D. Denneberg, Non-additive Measure and Integral, 2nd ed, Kluwer, Dordrecht, 1997.
[2] I. Dobrakov, On submeasures I, Dissertationes Math. 112 (1974) 1-35.
[3] I. Dobrakov, J. Farkova, On submeasures II, Math. Slovaca 30 (1980) 65-81.
[4] L. Drewnowski, Topological rings of sets, continuous set function, integration II, Bull. Acad.
Polon. Sci. Se. Sci. Math., Astr. Phys. 20 (1972) 277-286.
[5] Q. Jiang, G. J. Klir, Z. Wang, Null-additive fuzzy measures on -compact spaces, Int. J.
General Systems 25 (1993) 219-228.
$\mathrm{S}$[6] J. Li, Order continuous of monotone set function and convergence of measurable function
sequence, Appl. Math. Comput. 135 (2003) 211-218.

[7] J. Li, On Egoroff theorems on fuzzy measure spaces, Fuzzy Sets and Systems 135 (2003)

367-375.
[8] T. Murofushi, K. Uchino, S. Asahina, Conditions for Egoroff theorem in non-additive measure
theory, Fuzzy Sets and Systems 146 (2004) 135-146

10[9] K. Uchino, T. Murofushi, Relations between mathematical properties of fuzzy measures, Proc.
10th IFSA World Congr, Istanbul, Turkey (2003) 27-30.
[10] K. Uchino, S. Asahina, T. Murofushi, Dense pack property and null-continuity of fuzzy measure, in preparation.

[11] Z. Wang, The autocontinuity of set function and the fuzzy integral, J. Math. Anal AppL 99(1984) 195-218.
[12] Z. Wang, G.J. Klir Fuzzy Measure Theory, Plenum, New York, 1992