Journal article
On the upper and lower semicontinuity of the Aumann integral
Journal of mathematical economics, Vol.19(4), pp.373-389
01/01/1990
DOI: 10.1016/0304-4068(90)90028-8
Abstract
Let (
T,τ,μ) be a finite measure space,
X be a Banach space,
P be a metric space and let
L
1(μ,
X) denote the space of equivalence classes of
X-valued Bochner integrable functions on (
T,τ,μ). We show that if φ:
T×
P→2
X
is a set-valued function such that for each fixed
pϵ
P, φ(·,
p) has a measurable graph and for each fixed
tϵ
T, φ(
t,·) is either upper or lower semicontinuous then the
Aumann integral of φ, i.e.,∫
T
φ(
t,
p)d
μ(
t)= {∫
T
x(
t)d
μ(
t):
xϵS
φ
(
p)}, where
S
φ
(
p)= {
yϵL
1(
μ,
X):
y(
t)
ϵφ(
t,
p)
μ−a.e.}, is either upper or lower semicontinuous in the variable
p as well. Our results generalize those of Aumann (1965, 1976) who has considered the above problem for
X=
R
n
, and they have useful applications in general equilibrium and game theory.
Details
- Title: Subtitle
- On the upper and lower semicontinuity of the Aumann integral
- Creators
- Nicholas C. Yannelis - University of Illinois Urbana-Champaign
- Resource Type
- Journal article
- Publication Details
- Journal of mathematical economics, Vol.19(4), pp.373-389
- Publisher
- Elsevier B.V
- DOI
- 10.1016/0304-4068(90)90028-8
- ISSN
- 0304-4068
- eISSN
- 1873-1538
- Language
- English
- Date published
- 01/01/1990
- Academic Unit
- Economics
- Record Identifier
- 9984380407702771
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