In most economics textbooks there is a gap between the non-existence of utility functions and the existence of continuous utility functions, although upper semi-continuity is sufficient for many purposes. Starting from a simple constructive approach for countable domains and combining this with basic measure theory, we obtain necessary and sufficient conditions for the existence of upper semi-continuous utility functions on a wide class of domains. Although links between utility theory and measure theory have been pointed out before, to the best of our knowledge this is the first time that the present route has been taken.preferences, utility theory, measure theory, outer measure

Jörgen Weibull

Mark Voorneveld

English

Research Papers in Economics

 
 
OUTER MEASURE AND UTILITY 
 
 
Mark VOORNEVELD 
Jörgen W. WEIBULL 
 
 
October 2008 
 
 
Cahier n° 2008-28 
 
 
 
                              ECOLE POLYTECHNIQUE                       
CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE 
 
 
DEPARTEMENT D'ECONOMIE 
Route de Saclay 
91128 PALAISEAU CEDEX 
(33) 1 69333033 
http://www.enseignement.polytechnique.fr/economie/  
mailto:chantal.poujouly@polytechnique.edu 
 
 
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OUTER MEASURE AND UTILITY  
 
 
 
Mark VOORNEVELD
1 
Jörgen W. WEIBULL
2  
 
 
October 2008 
 
 
Cahier n° 2008-28 
 
 
 
 
Abstract:  In most economics textbooks there is a gap between the non-existence of utility functions and 
the existence of continuous utility functions, although upper semi-continuity is sufficient for 
many purposes. Starting from a simple constructive approach for countable domains and 
combining this with basic measure theory, we obtain necessary and sufficient conditions for 
the existence of upper semi-continuous utility functions on a wide class of domains. Although 
links between utility theory and measure theory have been pointed out before, to the best of 
our knowledge this is the first time that the present route has been taken. 
 
  
 
 
Key Words :   preferences, utility theory, measure theory, outer measure 
 
 
JEL Classification :  C60, D01 
 
 
   
 
 
                                                           
1 Stockholm School of Economics 
2 Stockholm School of Economics, Department of Economics Ecole Polytechnique 
We are grateful to Avinash Dixit, Klaus Ritzberger, and Peter Wakker for comments and to the Knut and Alice Wallenberg 
Foundation for financial support. 
 
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91. Introduction
In most economics textbooks there is a gap between the potential non-existence of utility func-
tions for complete and transitive preference relations on non-trivial connected Euclidean do-
mains — usually illustrated by lexicographic preferences (Debreu, 1954) — and the existence
of continuous utility functions for complete, transitive and continuous preferences on connected
Euclidean domains; see, e.g. Mas-Colell, Whinston, and Green (1995). Yet, for many purposes,
in particular for the existence of a best alternative in a compact set of alternatives, a weaker
property — upper semi-continuity — suﬃces. Hence, the reader of such a textbook treatment
might wonder if there exist upper semi-continuous utility functions, and whether this is true
even if the domain is not connected. We here ﬁll this gap providing necessary and suﬃcient
conditions for the existence of upper semi-continuous utility functions on arbitrary domains; see
Theorem 3.1 and Remark 3.2. Our approach is intuitive and constructive: we start by con-
structing utilities on a countable domain and then generalize the approach to arbitrary domains
by way of basic measure theory.
Measure theory is the branch of mathematics that deals with the question of how to deﬁne
the “size” (area/volume) of sets. We here formalize a direct intuitive link with utility theory:
given a binary preference relation on a set of alternatives, the “better” an alternative is, the
“larger” is its set of worse alternatives. So if one can measure the “size” of the set of worse
elements, for each given alternative, one obtains a utility function.
To be a bit more precise, measure theory starts out by ﬁrst deﬁning the “size” — measure —
of a class of “simple” sets, such as bounded intervals on the real line or rectangles in the plane,
and then extends this deﬁnition to other sets by way of approximation in terms of simple sets.
The outer measure is the best such approximation “from above”. This is illustrated in Figure 1,
where a set S in the plane is covered by rectangles. The outer measure S is the inﬁmum, over
S S
Figure 1: A set S and an approximation of its size using a covering.
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9all coverings by a countable number of rectangles, of the sum of the rectangles’ areas. In more
general settings, the outer measure is deﬁned likewise as the inﬁmum over coverings whose sizes
have been deﬁned; see, for instance, Rudin (1976, p. 304), Royden (1988, Sec. 3.2), Billingsley
(1995, Sec. 3), Ash (2000, p. 14).
We follow this approach by way of deﬁning the utility of an alternative as the outer measure
of its set of worse alternatives. We start by doing this for a countable set of alternatives, where
this is relatively simple and then proceed to arbitrary sets.
Our paper is not the ﬁrst to use tools from measure theory to address the question of utility
representation: pioneering papers are Neuefeind (1972) and Sondermann (1980). See Bridges
and Mehta (1995, sections 2.2 and 4.3) for a textbook treatment. However, our approach
diﬀers fundamentally from these precursors. Firstly, we only use the basic notion of outer
measure, while the mentioned studies impose additional topological and/or measure-theoretic
constraints.2 To the best of our knowledge, the logical connection between outer measure and
utility has never been made before. This link between utility theory and measure theory is
more explicit, intuitive and mathematically elementary than the above-mentioned approaches.
Secondly, our approach applies to a wider range of domains, in fact to all preference relations
that admit real-valued representations.
The rest of the paper is organized as follows. Section 2 recalls deﬁnitions and provides
notation. Our general representation theorem is given in Section 3. Its proof is in the appendix.
2. Deﬁnitions and notation
Preferences. Let preferences on an arbitrary set X be deﬁned in terms of a binary relation %
(“weakly preferred to”) which is:
complete: for all x,y ∈ X : x % y,y % x, or both;
transitive: for all x,y,z ∈ X: if x % y and y % z, then x % z.
As usual, x ≻ y means x % y, but not y % x, whereas x ∼ y means that both x % y and y % x.
The sets of elements strictly worse and strictly better than y ∈ X are denoted
W(y) = {x ∈ X : x ≺ y} and B(y) = {x ∈ X : x ≻ y}.
2Neuefeind (1972) restricts attention to ﬁnite-dimensional Euclidean spaces and assumes that indiﬀerence sets
have Lebesgue measure zero. Sondermann (1980) assumes that preferences are deﬁned on a probability space or
a second countable topological space; see also Corollary 3.4 below.
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9For x,y ∈ X with x ≺ y, the “open interval” of alternatives better than x but worse than y is
denoted
(x,y) = {z ∈ X : x ≺ z ≺ y}.
Topology. Given a topology on X, preferences % are:
continuous if for each y ∈ X, W(y) and B(y) are open;
upper semi-continuous (usc) if for each y ∈ X, W(y) is open.
Similarly, a function u : X → R is usc if for each r ∈ R, {x ∈ X : u(x) < r} is open.
Three important topologies are, ﬁrstly, the order topology, generated by (i.e., the smallest
topology containing) the collections {W(y) : y ∈ X} and {B(y) : y ∈ X}; secondly, the lower
order topology, generated by the collection {W(y) : y ∈ X}, and thirdly, for any subset D ⊆ X,
the D-lower order topology, generated by the collection {W(y) : y ∈ D}. By deﬁnition, the
order topology is the coarsest topology in which % is continuous; the lower order topology is the
coarsest topology in which % is usc.
As mentioned in the introduction, although one often appeals to continuity to establish
existence of most preferred alternatives, the weaker requirement of upper semi-continuity suﬃces.
A short proof: consider a complete, transitive, usc binary relation % over a compact set X. If
X has no most preferred element, then for each x ∈ X, there is a y ∈ X with y ≻ x, i.e., the
collection {W(y) : y ∈ X} is a covering of X with (by usc) open sets. By compactness, there are
ﬁnitely many y1,...,yk ∈ X such that W(y1),...,W(yk) cover X. Let yj be the most preferred
element of {y1,...,yk}. Then W(yj) covers the entire set X, a contradiction.
Utility. A preference relation % is represented by a utility function u : X → R if
∀x,y ∈ X :



x ∼ y ⇒ u(x) = u(y),
x ≻ y ⇒ u(x) > u(y).
(1)
3. Upper semi-continuous utility via outer measures
A complete, transitive binary relation % on a set X can be represented by a utility function if
and only if it is Jaﬀray order separable3 (Jaﬀray, 1975): there is a countable set D ⊆ X such
that for all x,y ∈ X:
x ≻ y ⇒ ∃d,d′ ∈ D : x % d ≻ d′ % y. (2)
3See Fishburn (1970, Sec. 3.1) or Bridges and Mehta (1995, Sec. 1.4) for alternative separability conditions.
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9Roughly speaking, countably many alternatives suﬃce to keep all pairs x,y ∈ X with x ≻ y
apart: x lies on one side of d and d′, whereas y lies on the other. To make our search for a (usc)
utility representation at all meaningful, we will henceforth focus on preference relations that are
Jaﬀray order separable.
The set D in the deﬁnition of Jaﬀray order separability is countable, so let n : D → N be an
injection. Finding a utility function on D is easy. Give each element d of D a positive weight
such that the weights have a ﬁnite sum and use the total weight of the elements weakly worse
than d as the utility of d. For instance, give weight 1
2 to the alternative d with label n(d) = 1,
weight 1
4 to the alternative d with label n(d) = 2, and inductively, weight w(d) = 2−k to the
alternative d with label n(d) = k. In general, let (εk)∞
k=1 be a summable sequence of positive
weights; w.l.o.g. its sum
 ∞
k=1 εk is one. Assign to each d ∈ D weight w(d) = εn(d).4 Deﬁne
u0 : D → R for each d ∈ D by u0(d) =
 
d′-d w(d′). Clearly, (1) is satisﬁed.
We can extend this procedure from D to X as follows. Let W = {W(d) : d ∈ D} ∪ {∅,X}
and deﬁne ρ : W → [0,1] as follows: ρ(∅) = 0,ρ(X) = 1 and for d ∈ D:
ρ(W(d)) =
 
d′∈D:d′-d
w(d′). (3)
Notice that W is countable and that it is a covering of X. Extend ρ to an outer measure  ∗ on
X in the usual way (recall Figure 1): for each set A ⊆ X, deﬁne  ∗(A) as the smallest total size
of sets in W covering A:
 ∗(A) = inf{
 
i∈N ρ(Wi) : {Wi}i∈N ⊆ W,A ⊆ ∪i∈NWi}.
Deﬁne u : X → R for each x ∈ X as the outer measure of the set of elements worse than x:
u(x) =  ∗(W(x)). (4)
This outer measure gives the desired utility representation:
Theorem 3.1 Consider a complete, transitive, Jaﬀray order separable binary relation % on an
arbitrary set X. The outer-measure utility function u in (4) represents % and is usc in the
D-lower order topology.
4 If there is a worst element in X (an x
0 ∈ X with x
0 - x for all x ∈ X), one may assume w.l.o.g. that
D contains one such element, say d. Its weight can be normalized to zero: w(d) = 0. This will assure that
ρ(W(d)) = ρ(∅) = 0 in (3).
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9Remark 3.2 We chose to formulate the theorem in its present form, as it stresses that the
outer-measure utility function represents preferences whenever a utility representation is at all
feasible, even if a usc representation is impossible. Yet it gives necessary and suﬃcient conditions
for representation in terms of a usc utility function: there is a usc utility function representing
preferences % if and only if they are complete, transitive, Jaﬀray order separable, and usc in
any topology equal to or ﬁner than the D-lower order topology. ⊳
Corollaries 3.3 and 3.4 below provide applications of our general theorem. Consider preferences
% over a commodity space X = Rn
+ (n ∈ N) with its standard Euclidean topology.5 If % is
usc in this topology, it is Jaﬀray order separable (Rader, 1963). By assumption, W(y) is open
for each y ∈ Rn
+, so the Euclidean topology is ﬁner than the D-lower order topology. Hence,
Theorem 3.1 applies:
Corollary 3.3 If % is a complete, transitive binary relation over Rn
+ (n ∈ N) and usc in the
Euclidean topology, the outer-measure utility function in (4) represents % and is usc in the
Euclidean topology.
The closest result we could ﬁnd to Theorem 3.1 is a result in Sondermann (1980), which is a
special case. Call a preference relation % on a set X perfectly separable if there is a countable
set C ⊆ X such that for all x,y ∈ X, with x  ∼ c and y  ∼ c for all c ∈ C, the following holds:
x ≻ y ⇒ ∃c ∈ C : x ≻ c ≻ y.
Perfect separability implies Jaﬀray order separability (Jaﬀray, 1975), so:
Corollary 3.4 [Sondermann, 1980, Corollary 2] Consider a complete, transitive, perfectly
separable binary relation % on a set X. Then there is a utility function representing %, usc in
any topology equal to or ﬁner than the lower order topology.
Appendix: Proof of Theorem 3.1
Preliminaries. By deﬁnition,
∀d ∈ D : u(d) =  ∗(W(d)) = ρ(W(d)) =
 
d′∈D:d′-d
w(d′), (5)
5Or any other topological space with a countable base.
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9and the outer measure  ∗ is monotonic: if A ⊆ B ⊆ X, then  ∗(A) ≤  ∗(B).
Representation. We prove (1). Let x,y ∈ X. If x ∼ y, then W(x) = W(y) by transitivity of
%, so u(x) = u(y). If x ≻ y, there are d,d′ ∈ D with x % d ≻ d′ % y by (2). By monotonicity of
 ∗ and (5): u(x) =  ∗(W(x)) ≥  ∗(W(d)) >  ∗(W(d′)) ≥  ∗(W(y)) = u(y).
Semi-continuity. Let r ∈ R. We show that {x ∈ X : u(x) < r} is open. To avoid trivialities,
let {x ∈ X : u(x) < r}  = ∅,X. Hence, there is a y∗ ∈ X with r ≤ u(y∗) ≤ 1. Let x ∈ X have
u(x) < r. It suﬃces to show that there is an open neighborhood V of x with u(v) < r for each
v ∈ V .
Case 1: There is no d ∈ D with d ∼ x. As D may be assumed to contain a worst element of X, if
such exists (see footnote 4): W(x)  = ∅. By deﬁnition of  ∗, there are {Wi}i∈N ⊆ W with W(x) ⊆
∪i∈NWi and  ∗(W(x)) ≤
 
i∈N ρ(Wi) < r ≤ 1. As W(x)  = ∅, the set J = {i ∈ N : Wi  = ∅} is
nonempty. As ρ(X) = 1 and
 
i∈N ρ(Wi) < 1, Wi  = X for each i ∈ J. So for each i ∈ J there is
a di ∈ D with Wi = W(di). We show that di ≻ x for some i ∈ J. Suppose, to the contrary, that
di ≺ x for each i ∈ J. For each j ∈ J, the set {di ∈ D : i ∈ J,di % dj} is inﬁnite: otherwise, it
has a best element d∗, but then ∪i∈NWi = ∪i∈JW(di) = W(d∗) is a proper subset of W(x) by
Jaﬀray order separability, contradicting W(x) ⊆ ∪i∈NWi. Let j ∈ J with ρ(W(dj)) := ε > 0. By
the above, there are inﬁnitely many i ∈ J with ρ(Wi) = ρ(W(di)) ≥ ρ(W(dj)) = ε, contradicting
that
 
i∈N ρ(Wi) < 1. We conclude that di ≻ x for some i ∈ J. So x ∈ W(di), an open set in
the D-lower order topology, and for each v ∈ W(di): u(v) < u(di) = ρ(W(di)) < r.
Case 2: There is a d ∈ D with d ∼ x. By (2) and y∗ ≻ x: B(d) ∩ D = {d′ ∈ D : d′ ≻ d}  = ∅.
Case 2A: There is a d′ ∈ B(d) ∩ D with (d,d′) = ∅. Then {z ∈ X : z - d} = {z ∈ X : z ≺
d′} = W(d′) is open in the D-lower order topology, contains x, and for each z ∈ W(d′) : u(z) ≤
u(d) = u(x) < r.
Case 2B: For each d′ ∈ B(d) ∩ D, (d,d′)  = ∅. Then by (2), there is, for each d′ ∈ B(d) ∩ D,
a d′′ ∈ B(d) ∩ D that is strictly worse: d′′ ≺ d′. So B(d) ∩ D is inﬁnite. Since the sequence of
weights (εk)∞
k=1 is summable, there is a k ∈ N such that
 ∞
ℓ=k εℓ < r − u(x). Since there are
only ﬁnitely many d′ ∈ D with n(d′) < k, there is a suﬃciently bad d∗ ∈ B(d) ∩ D such that
n(d′) ≥ k for each d′ ∈ B(d) ∩ D with d′ - d∗.
Since d∗ ∈ B(d) ∩ D, x ∈ W(d∗), which is open in the D-lower order topology. By x ∼ d
and the construction of d∗:
u(x) =
 
d′∈D:d′-d
w(d′)
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9and
 
d′∈B(d)∩D:d′-d∗
w(d′) =
 
d′∈B(d)∩D:d′-d∗
εn(d′) ≤
∞  
ℓ=k
εℓ < r − u(x).
Hence, for each v ∈ W(d∗):
u(v) < u(d∗) = ρ(W(d∗)) =
 
d′∈D:d′-d
w(d′) +
 
d′∈B(d)∩D:d′-d∗
w(d′) < u(x) + r − u(x) = r.
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Existence of a continuous utility function: An elementary proof.

Microeconomic Theory.

On continuous utility.

Probability and Measure Theory, 2nd edition. London/San Diego:

Probability and Measure, 3rd edition.

Real Analysis, 3rd edition.

Representation of a preference ordering by a numerical function. In: Decision Processes.

Outer measure and utility

Abstract

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