The existence of a purely finitely additive measure cannot be proved in Zermelo-Frankel set theory if the use of the Axiom of Choice is disallowed.Finitely additive probabilities; Charges; Axiom of choice; Constructivism.

Luc LAUWERS

English

Research Papers in Economics

 
Purely finitely additive measures are non-constructible objects 
 
by 
 
 
Luc LAUWERS 
 
Econometrics 
 
 
Center for Economic Studies 
Discussions Paper Series (DPS) 10.10 
http://www.econ.kuleuven.be/ces/discussionpapers/default.htm 
 
 
April 2010 Purely nitely additive measures are non-constructible
objects
Luc Lauwers
Center for Economic Studies, K.U.Leuven
Naamsestraat 69, B-3000 Leuven, BELGIUM
Luc.Lauwers@econ.kuleuven.be
April 13, 2010
Abstract. The existence of a purely nitely additive measure cannot be proved
in Zermelo-Frankel set theory if the use of the Axiom of Choice is disallowed.
Key words. Finitely additive probabilities; Charges; Axiom of choice; Con-
structivism.
1 Introduction
A nitely additive measure  on N assigns to each subset of N a nonnegative real number
and assigns to the union of disjoint sets the sum of their numbers. The measure  is said to
be countably additive if the measure of a countable union of pairwise disjoint sets is equal
to the sum of the measures of those sets. The nitely additive measure  is dominated
by  (and we write   ) is for each subset S of N, we have (S)  (S). The nitely
additive measure  is said to be purely nitely additive if the inequalities 0     with
 countably additive imply that  = 0. From Yosida and Hewitt (1952) and Rao (1958) we
know that each nitely additive measure uniquely decomposes as the sum of a countably
additive and a purely additive measure. Typically, a purely nitely additive measure is
obtained by means of Hahn-Banach's theorem or by means of a free ultralter.1 Let us
describe the second route.
This note extends the result obtained in \The uniform distributions puzzle" (2007, 2009) to arbitrary
purely nitely additive measures. I thank professors J.B. Kadane, Philippe Mongin, Stephen Portnoy,
K.P.S. Bhaskara Rao, and T. Seidenfeld for their comments on these earlier versions.
1A free ultralter on the set N of natural numbers is a collection U of subsets of N such that (i) N 2 U
and ? = 2 U, (ii) if A  B and A 2 U, then B 2 U, (iii) if A;B 2 U, then A \ B 2 U, and (iv) for each
A  N, either A or N   A belongs to U. The existence of a free ultralter follows from Zorn's Lemma.
1A free ultralter F on N denes a limit on X. Consider a sequence x in X and all of its
limit points. Each limit point is the limit of a subsequence. There is only one limit point
with a converging subsequence xi1;xi2;:::;xit;::: for which the set fi1;i2;:::;it;:::g of
indices belongs to F. Dene limF(x) = limt!1 xit. Due to the intersection property of F,
we have limF(x+y) = limF(x)+limF(y) for each x and y in X. The ultralter-based-limit
limF denes a nitely additive measure:
F(S) = lim
F
st with st =
#(S \ f1;2;:::;tg)
t
;
and S a subset of N. If the sequence s1;s2;:::;st;::: has only one accumulation point,
then F(S) coincides with `the' limit of this sequence and is known as the natural density
of S. For example, the set of even numbers has a natural density equal to .5; the set of
all multiples of 20 has a natural density equal to .05. Not every subset of N has a natural
density. For example, the set
S1 = f1;10;11;:::;19;100;101;:::;199;1000;1001;:::g
of all natural numbers having their rst digit equal to 1 has no natural density. The
measure F(S1) depends upon the particular (non-constructible!) ultralter F and can
take any value between 1/9 and 5/9.2
Both routes to obtain purely nitely additive measures (Hahn-Banach's theorem and a
free ultralter) rely upon AC and involve non-constructive methods. Obviously, one cannot
conclude from this that purely nitely additive measures are non-constructible objects. The
knowledge that non-constructive methods can be used to obtain a purely nitely additive
measure, does not answer the question whether a purely nitely additive measure can be
obtained without recurse to non-constructive methods.
This note shows that the existence of a purely nitely additive measure on N entails
the existence of a non-Ramsey set. From Mathias (1977) we know that a non-Ramsey set
is a non-constructible object.
The next section touches the notion of constructivism and recalls the Axiom of Choice
and the Axiom of Dependent choice. Section 3 states and proves the main result. The
result extends to purely nitely measures on R.
2 Constructivism
The Axiom of Choice (AC) postulates for each nonempty family D of nonempty sets the
existence of a function f such that f(S) 2 S for each set S in the family D. The function
f is referred to as a choice function. AC does not provide an explicit way to construct
such a choice function and provoked considerable criticism in the aftermath of Zermelo's
2In this example, liminf(st) is the limit of the sequence 1=9, 11=99, 111=999;::: and is equal to 1/9;
limsup(st) is the limit of the sequence 1, 11=19, 111=199;::: and is equal to 5/9.
2formulation in 1904.3 Among the applications of AC, we mention Zorn's Lemma, the
theorem of Hahn-Banach, and the existence of free ultralters. AC also implies a number
of paradoxes such as the decomposition of a sphere into a sphere of smaller size, and
the existence of a non-measurable set of real numbers. The nonconstructive character of
AC is further revealed by Dianonescu (1975) who showed that AC implies the law of the
excluded middle.4 Constructive mathematics rejects the law of the excluded middle and
hence rejects AC. On the other hand, the Axiom of Dependent Choice (DC) is generally
accepted by constructivists (Beeson, 1988, p. 42). Let S be a nonempty set and let R be
a binary relation in S such that for each a in S there is a b in S with (a;b) 2 R. Then,
DC postulates the existence of a sequence (a1;a2;:::;an;:::) of elements in S such that
(ak;ak+1) 2 R for each k = 1;2;:::.
The nonconstructive object used in this note is known as a non-Ramsey set. Let I be
an innite set and let n be a positive integer. Let [I]n collect all the subsets of I with
exactly n elements. Ramsey (1928) shows that for each subset S of [I]n, there exists an
innite set J  I such that either [J]n  S or [J]n \ S = ?. When n is replaced by
countable innity, then Ramsey's theorem fails. There exists a subset S of [I]1 such that
for each innite subset J of I the class [J]1 intersects S and its complement [I]1   S as
well. Such a set S is said to be non-Ramsey. Mathias (1977) showed that the existence of
non-Ramsey sets does not follow from ZF (without AC).5
3 Finitely additive measures
Let N be the set of natural numbers. Let F be the eld of all subsets of N. A nitely
additive probability is a map  : F ! R+ that satises and the condition


A1 [ A2 [ ::: [ An

= (A1) + (A2) +  + (An);
where the sets A1;A2;:::;An are disjoint, for all nite n. We now state the main result of
this note.
Theorem. The existence of a nitely additive measure  on N that attaches zero probability
to each natural number entails the existence of a non-Ramsey set.
Proof. Rescale the measure  such that (N) = 1. We use some additional notation. For
two natural numbers i > j, let [i;j[ denote the set fi;i+1;:::;j 1g. Furthermore, to each
3AC is (i) consistent and (ii) independent: (i) AC can be added to the Zermelo-Fraenkel axioms of set
theory (ZF) without yielding a contradiction, and (ii) AC is not a theorem of ZF (Fraenkel et al, 1973).
4The law of the excluded middle states the truth of `P or not-P' for each proposition P and can be
used to claim the existence of certain objects without any hint to its construction. For example, the real
number c =
p
2
p
2 either is rational (in which case one sets a = b =
p
2) or is not rational (in which case
one sets a = c and b =
p
2). Conclude the existence of irrational numbers a and b for which ab is rational.
5More precisely, Solovay (1970) proposed a model in which ZF and DC are true and in which AC fails.
Mathias showed that in this Solovay-model a non-Ramsey set does not exist. Hence, the existence of a
non-Ramsey set is independent of ZF + DC.
3innite set A  N we connect a set, denoted by A0, as follows. Let A = fn0;n1;:::;nk;:::g
with nk < nk+1 for each k, then A0 = [n0;n1[ [ [n2;n3[ [ ::: [ [n2k;n2k+1[ [ ::::
Now, let  satisfy the requirements listed in the theorem. We show that
S = fA  Nj(A0) > 0:5g
is a non-Ramsey set. It is sucient to show that each innite set A = fn0;n1;:::;nk;:::g
includes an innite subset B such either A or B belongs to S (the `either-or' being exclu-
sive). We distinguish three cases.
Case 1. A = 2 S, in particular (A0) < 0:5. Let B = A fn0g. Then, [0;n0[[A0[B0 = N.
Since ([0;n0[) = 0, (A0) < 0:5, and (N) = 1; we obtain that (B0) > 0:5. Therefore,
B  A and B 2 S.
Case 2. A = 2 S, in particular (A0) = 0:5. Let B = fn0;n3;n4;n7;:::;n4k;n4k+3;:::g and
let B0 = fn0;n1;n2;n5;n6;n9;:::;n4k+2;n4k+5;:::g. Then, A0 = B0 \ B0
0. Hence, we have
(B0)  0:5 and (B0
0)  0:5. Furthermore,
[0;n0[ [ (B0 B
0
0) [ A0 = N:
Conclude that the symmetric dierence B0 B0
0 has a measure equal to 0.5. Hence, at
least one of the sets B0 or B0
0 has a measure strictly larger than 0.5. Select the subset B
or B0 of A for which the corresponding set B0 or B0
0 has the highest measure. The selected
subset of A belongs to S.
Case 3. A 2 S. Similar to Case 1, we put B = A   fn0g. Conclude that B  A and that
B = 2 S. 2
Finally, we indicate how this result extends to purely nitely additive measures on R.
Consider such a measure . Since,  is not countably additive, there exists a countable
sequence of pairwise disjoint sets A1, A2;:::, An;::: (subsets of R) such that
(A1) + (A2) +  + (An) +  < (A1 [ A2 [  [ An [ ):
Dene a measure 0 on N by 0(C) = ([j2CAj). This measure has a purely nitely
additive component.
References
Beeson MJ (1988) Foundations of constructive mathematics. Ergebnisse der Mathematik
und ihrer Grenzgebiete 3.6. Berlin: Springer-Verlag.
Bhaskara Rao KPS, Bhaskara Rao M (1983) Theory of charges, a study of nitely additive
measures. London: Academic Press.
Chichilnisky G (1998) The economics of global environmental risks. International Year-
book of Environmental and Resource Economics, Vol. II eds. T Tietenberg and H
Folmer, Edward Elgar, 235-273.
4Chichilnisky G (2000) An axiomatic approach to choice under uncertainty with catas-
trophic risks. Resource and Energy Economics 22(3) 221 -231.
Chichilnisky G (2009) Avoiding extinction: equal treatment of the present and the future.
Economics: The Open-Access, Open-Assessment E-Journal 3, 2009-32.
http://www.economics-ejournal.org/economics/journalarticles/2009-32.
Chichilnisky G (2009) Subjective Probability with Black Swans. Journal of Probability
and Statistics, Special Issue on Actuarial and Financial Risks (in press).
Chichilnisky G (2009) The Foundations of Statistics with Black Swans. Mathematical
Social Sciences, doi:10.1016/j.mathsocsci.2009.09.007.
Chichilnisky G (2009) The Topology of Fear. Journal of Mathematical Economics 45
Issue 11-12.
Chichilnisky G (2009) The limits of econometrics: nonparametric estimation in Hilbert
spaces. Econometric Theory 25(04), 1070-1086.
Chichilnisky G (2009) Avoiding Extinction: Equal Treatment of the Present and the
Future. Economics: The Open-Access, Open-Assessment E-Journal, 2009, Vol. 3,
2009-32.
Deal RB, Waterhouse WC, Alder RL, Konheim AG, and Blau JG (1963) Probability mea-
sure for sets of positive integers. The American Mathematical Monthly (Advanced
Problems and Solutions) 70(2), 218-219.
Diaconis P (1977) The distribution of leading digits and uniform distribution mod 1. The
Annals of Probability 5(1), 72-81.
Dubins LE, Savage LJ (1976) Inequalities for stochastic processes, how to gamble if you
must. New York: Dover Publications.
Fishburn P (1986) The axioms of subjective probability. Statistical Science 1(3), 335-358.
Fraenkel AA, Bar-Hillel Y, and Levy A (1973) Foundations of set theory. Amsterdam:
North-Holland.
Kadane JB, O'Hagan A (1995) Using nitely additive probability: uniform distributions
on the natural numbers. Journal of the American Statistical Association 90, 626-631.
Kadane JB, Schervish M, and Seidenfeld T (1986) Statistical implications of nitely ad-
ditive probability. p59-76 in Bayesian Inference and Decision Techniques: Essays in
Honor of Bruno de Finetti, edited by PK Goel and A Zellner, Amsterdam: North
Holland.
Kadane JB, Schervish M, and Seidenfeld T (1999) Rethinking the foundations of statistics.
Cambridge UP, 1999.
5Mathias ARD (1977) Happy families. Annals of Pure and Applied Logic 12, 59-111.
Ramsey FP (1928) On a problem of formal logic. Proceedings of the London Mathematical
Society, Series 2. 30.4, 338-384.
Rao RR (1958) A note on nitely additive measures. The Indian Journal of Statistics 19,
27-28.
Savage LJ (1954) The foundations of statistics. New York: Dover Publications, 2nd edn,
1972.
Seidenfeld T, Schervish M (1983) A conict between nite additivity and avoiding Dutch
book. Philosophy of Science 50, 1983, 398-412.
Schirokauer O, Kadane JB (2007) Uniform Distributions on the Natural Numbers Journal
of Theoretical Probability (2007) 20, 429-441.
Sierpi nski W (1938) Fonctions additives non compl etement additives et fonctions non
mesurables. Fundamenta Mathematicae 30, 96-99.
Solovay R (1970) A model of set theory where every set of reals is Lebesgue measurable.
Annals of Mathematics 92, 1-56.
Stinchcombe MB (1997) Countably Additive Subjective Probabilities. Review of Eco-
nomic Studies 64(1), 125-146.
Wakker P (1993) Savage's axioms usually imply violation of strict stochastic dominance.
Review of Economic Studies 60, 487-493.
Yosida K, Hewitt E (1952) Finitely additive measures. Transactions of the American
Mathematical Society 72, 46-66.
6Copyright © 2010 @ the author(s). Discussion papers are in draft form. This discussion paper 
is distributed for purposes of comment and discussion only. It may not be reproduced without 
permission of the copyright holder. Copies of working papers are available from the author. 
 
 Copyright © 2010 @ the author(s). Discussion papers are in draft form. This discussion paper 
is distributed for purposes of comment and discussion only. It may not be reproduced without 
permission of the copyright holder. Copies of working papers are available from the author. 
 
 

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Purely finitely additive measures are non-constructible objects

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