Non-Additive Measure and Integral

181 Pages·1994·7.32 MB·English

by  Dieter Denneberg (auth.)

NON-ADDITIVE MEASURE AND INTEGRAL
THEORY AND DECISION LIBRARY
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VOLUME 27
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NON-ADDITIVE MEASURE
AND INTEGRAL
by
DIETER DENNEBERG
Universitiit Bremen
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Denneberg. Dieter.
Non-additive measure and integral I by Dieter Denneberg.
p. cm. -- (Theory and decision I ibrary. Series B.
Mathematical and statistical methods; v. 27)
Includes bibliographical references and index.
ISBN 978-90-481-4404-4 ISBN 978-94-017-2434-0 (eBook)
DOI 10.1007/978-94-017-2434-0
1. Measure theory. 2. Integrals. Generalized. I. Title.
II. Series.
OA312.D36 1994
515' .42--dc20 94-1488-0
ISBN 978-90-481-4404-4
Printed on acidjree paper
All Rights Reserved
© 1994 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1994
Softcover reprint of the hardcover I st edition 1994
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Contents
Preface
Vll
1 Integration of Monotone Functions on Intervals 1
2 Set Functions and Caratheodory Measurability 15
3 Construction of Measures using Topology 35
4 Distribution Functions, Measurability and Comonotonicity
of Functions 45
5 The Asymmetric Integral 61
6 The Sub additivity Theorem 71
7 The Symmetric Integral 87
8 Sequences of Functions and Convergence Theorems 93
9 Nullfunctions and the Lebesgue Spaces Lp 103
10 Families of Measures and their Envelopes 123
11 Densities and the Radon-Nikodym Theorem 129
12 Products 145
13 Representing Functionals as Integrals 155
References 171
Index 175
VB
Lebesgue expliquait la nature de son integrale
par une image plaisante et accessible a tous.
"Je dois payer une certaine somme, disait-ilj je
fouille dans mes poches et j'en sors des pieces et
des billets de differentes valeurs. Je les verse a
mon creancier dans l'ordre ou elles se presentent
jusqu'a atteindre Ie total de rna dette. C'est
l'integrale de Riemann. Mais je peux operer
autrement. Ayant sorti tout mon argent, je
reunis les billets de meme valeur, les pieces sem
blables, et j'effectue Ie paiement en donnant en
semble les signes monetaires de meme valeur.
C'est mon integrale." (Denjoy, Felix, Montel
1957).
PREFACE
Reading the above quotation one might imagine that integration the
ory could be founded on order. Indeed it turned out that many aspects
of integration theory are sustained if additivity is replaced by order and
monotonicity. This approach depends essentially on the natural order
ing of the real numbers. Thus it does not generalize naturally to vector
valued functions as the additive theory does.
n
The key to the general integral of a real function with domain is
it's distribution function with respect to a "measure" or set function,
as we shall call it in general, on n. Under minimal requirements on
the set function, essentially monotonicity, the distribution function is a
monotone function on the real line and by means of usual (Riemann)
integration the general integral of the initial function is defined. The
general integral, of course, does not behave additive, but it remains
additive in a restricted sense, namely for comonotonic functions.
If, in applications, non-additive "measures" appear they often are
still subadditive or, more restrictive, submodular. In the latter case
many results of additive integration theory can be sustained: the integral
behaves subadditive and linear normed function spaces Lp , the Lebesgue
Vlll
spaces, can be defined.
The sub additivity theorem had been a first substantial contribution
to non-additive integration theory. It is due to Choquet 1953/54 who
was lead to the problem from his research in electrostatics (non-additive
measures are sometimes called capacities) and potential theory. Then
Choquet's results had been applied to statistics by Strassen, Huber and
others. In the context of statistics a recent development (Walley 1991)
even goes beyond our theory: the functionals studied there are still
sub additive and positively homogenous but not comonotonic additive,
hence they are no integral in our sense. In economic decision theory non
additive measure and integral appeared at first implicitly and indepen
dently in Quiggin 1982, Schmeidler 1986 and Yaari 1987. The last two
authors made clear the importance of comonotonicity. Further topics
with relations to non-additive measure and integral are belief functions
(developed by Dempster and Shafer), cooperative game theory, fuzzy
measures, artificial intelligence and many others. A purely mathemati
cal very general approach to the fundamentals of non-additive measure
and integral appeared in papers of Greco 1977, 1981, 1982 and Bas
sanezi and Greco 1984, written in Italian or French. Greco developed a
natural notion of measurability which we adapt in the present notes: A
n
function on is measurable with respect to a monotone set function if
its distribution functions with respect to all monotone extensions of the
n
set function to the whole power set of are identical. Altogether one
can observe that the fundamentals of non-additive integration had been
developed independently again and again. There is no unified, widely
known and accepted approach to the theory. We hope, the present notes
can contribute to this objective.
Our approach comprises finitely additive measures (which are called
charges in Bhaskara Rao and Bhaskara Rao 1983) and a-additive mea
sures and their respective integrals as special cases. There are never
theless some topics (Lebesgue-Stieltjes measures, Radon-Nikodym The
orem, Fubinis Theorem) where the results don't reach far beyond the
a-additive case. They had been included for the sake of completeness
and with the intention that the present notes could be employed for
courses on measure and integration where these important topics can
not be skipped. Conditional expectation, a likewise important issue, is
ix
not treated here since it is not yet clear how to generalize it for non
additive or at least submodular set functions. But see Denneberg 1994
and Chapter 12 for partial results.
In classical, a-additive measure and integration theory a fundamen
tal tool is to approximate a function by simple functions. In our ap
proach this method is partly replaced by approximating a set function
by simpler set functions. This method would not be sufficiently flexible
if it were restricted to additive set functions. Another typical exam
ple for what can be gained from greater flexibility in the set functions
is the identity Loo(J-l) = L1 (sign J-l) of Lebesgue spaces. Our approach
can be of interest, too, for the classical a-additive case. For example,
it reveals which phenomena are due to additivity and which to conti
nuity of the set function. Another point in which the present approach
distinguishes from other approaches is the consistent use of decreasing
distribution functions or their pseudo-inverse functions. Then, for exam
ple, the transformation formula for integrals is nearly trivial or issues as
stochastic convergence arise naturally in studying sequences of functions
and convergence theorems for integrals.
A more detailed information on the content and structure of the
present notes can be gained in reading the introductory overviews of the
single chapters. Exercises are appended to each chapter. Some of them
are elementary to help understanding the text, others are intended to
supplement the text. Exercises 2.11, 6.7, 11.3 hint at specific applica
tions of the theory to belief functions, welfare economics, decision theory
or insurance mathematics.
The author gratefully acknowledges detailed comments from A.Cha
teauneuf, D.Plachky and U.Wortmann. Also discussions with many col
leagues and students played an important role in the longlasting process
of writing the present notes. Among them I mention T .E.Armstrong,
G.Bamberg, M.Cohen, G.Debs, C.Dellacherie, R.Dyckerhoff, I. Gil
boa, J.Y.Jaffray, H.P.Kinder, M.Machina, K.Mosler, L.Riischendorf,
D.Schmeidler, P.Wakker, M.E.Yaari. Many versions of the manuscript
had been typed by H.Siebert and the final accomplishment was done by
C.Cebulla and T.Rathe.
Bremen, January 1994 Dieter Denneberg
Chapter 1
Integration of Monotone
Functions on Intervals
Our approach to the general theory of integration is based, via distribu
tion functions, upon the integral of monotone functions on intervals. The
latter is already provided by the (improper) Riemann integral. For the
sake of completeness, to fix the terminology and to prepare subsequent
proofs we survey integration of monotone functions. We are working
with countable subdivisions to include the improper Riemann integral
from the beginning. Crucial for later chapters will be the pseudo-inverse
function of a decreasing function. It is introduced in the present chapter.
Let iR:= IR U {-oo, oo} be the extended real line and IR+ :=
{x E iR I x 2:: O}. We extend the natural ordering of the reals through
-00 < r < 00 for r E IR and use the following conventions,
inf 0 00 = sup IR, inf IR -00 sup 0,
c·oo 00, c·(-oo) -00 for c > 0,
c·oo -00, c·(-oo) 00 for c < 0,
00 + r 00 for r > -00, -00 + r -00 for r < 00.
Other operations like 00 - 00 are not defined.
For a (weakly) decreasing function f : I iR on an (open, closed
---7
or semiclosed) interval I C iR and for a subdivision d : Z I of I
---7
= =
with dn ::; dn+I, n E Z, and inf dn inf I, sup dn sup I we define
nEZ nEZ
1
2 Non-additive Measure and Integral
the lower sum
L00
S(j,d):= f(dn)(dn-dn-d·
n=-oo
This sum may not be defined, namely if the positive summands sum up
to 00 and the negative summands to -00. Otherwise S (j, d) E ill. If
f 2: 0 we always have S (j, d) E iR+ . Now we define the integral of f
on I as J
f(x)dx := s~p S(f,d),
I
where the supremum extends over all subdivisions of I as above. If
S (j, d) does not exist for all d, then we define J f( x) dx as non existing.
Otherwise the integral is in ill. It can easily be checked that the value
of the integral is not affected if the boundary points of I are added to I
or are removed from I. Hence we write as usual
Jb J
f(x)dx:= f(x)dx if 1= ]a,b[, [a,b[, ]a,b] or [a,b].
a I
The following properties for decreasing functions f, 9 on I are easily
verified with methods familiar from the Riemann integral.
b
(i) J c dx = c (b - a) , c E IR, a ::; b .
a
=
(ii) Jcf(x)dx cJf(x)dx, c2:0 (cfisnotdecreasingifc<O).
I I
+ = +
(iii) J (j g) dx J f dx J 9 dx .
I I I
(iv) f::; 9 on I implies J f( x) dx ::; J g( x) dx .
I I
c b c
+
(v) J f( x) dx = J f( x) dx J f( x) dx , a::; b ::; c.
a a b
(vi) Invariance under translations of ill,
b+c b
J fey - c) dy = J f( x) dx , c E IR.
a