A generalization of Fatou’s lemma for extended realvalued functions on σfinite measure spaces: with an application to infinitehorizon optimization in discrete time
Kamihigashi Journal of Inequalities and Applications
A generalization of Fatou's lemma for extended realvalued functions on σ finite measure spaces: with an application to infinitehorizon optimization in discrete time
Takashi Kamihigashi
Given a sequence {fn}n∈N of measurable functions on a σ finite measure space such that the integral of each fn as well as that of lim supn↑∞ fn exists in R, we provide a sufficient condition for the following inequality to hold: lim sup n↑∞
Fatou's lemma; σ finite measure space; infinitehorizon optimization; hyperbolic discounting; existence of optimal paths

Our condition is considerably weaker than sufficient conditions known in the
literature such as uniform integrability (in the case of a finite measure) and
equiintegrability. As an application, we obtain a new result on the existence of an
optimal path for deterministic infinitehorizon optimization problems in discrete time.
1 Introduction
Some sufficient conditions for this inequality weaker than the one described above are
‘uniform integrability’ of {fn+} (where fn+ is the positive part of fn) is a sufficient condition for
the Fatou inequality (.) in the case of a finite measure (e.g., [–]); so is ‘equiintegrability’
precisely defined in Section .
© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium,
provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and
indicate if changes were made.
In this paper we provide a sufficient condition for the Fatou inequality (.) considerably
weaker than the above conditions. Our approach is based on the following assumption,
which is maintained throughout the paper.
Under this assumption there is an increasing sequence of measurable sets of finite
measure whose union equals . We use this sequence to specify a ‘direction’ in which we
successively approximate the integral of a function.
There is a natural increasing sequence of measurable sets if the measure space is the set
of nonnegative integers equipped with the counting measure. In this setting, we provide
a simple sufficient condition for the Fatou inequality (.) as a corollary of our general
result. Applying this condition to a fairly general class of infinitehorizon deterministic
optimization problems in discrete time, we establish a new result on the existence of an
optimal path. The condition takes a form similar to transversality conditions and other
related conditions in dynamic optimization (e.g., [–]).
The current line of research was initially motivated by the limitations of the existing
applications of Fatou’s lemma to dynamic optimization problems (e.g., [, ]). In
particular, there are certain cases in which optimal paths exist but the standard version of Fatou’s
lemma fails to apply. This is illustrated with some examples following our existence result.
We should mention that there are other important extensions of Fatou’s lemma to more
general functions and spaces (e.g., [–]). However, to our knowledge, there is no
result in the literature that covers our generalization of Fatou’s lemma, which is specific to
extended realvalued functions.
In the next section we define the concepts and conditions needed to state our main result
and to compare it with some previous results based on uniform integrability and
equiintegrability. In Section we state our main result and derive those previous results as
consequences. In Section we present two simple examples that cannot be treated by the
previous results but that can easily be treated using our result. In Section we show a new
result on the existence of an optimal path for infinitehorizon deterministic optimization
problems in discrete time. In Section we prove our main result.
2 Definitions
It is easy to see that μ is σ finite if and only if there exists a σ finite exhausting sequence.
Since we assume that μ is σ finite, we have at least one σ finite exhausting sequence.
A sequence {fn}n∈N of integrable functions in L( ) is called equiintegrable (e.g., [],
page ) if the following conditions hold:
Ai ⊂ Ai+,
Ai = .
i∈N
Suppose that μ( ) < ∞. A sequence {fn}n∈N of integrable functions in L( ) is called
uniformly integrable (e.g., [], page ) if
It is well known that a sequence {fn}n∈N of integrable functions in L( ) is uniformly
integrable if and only if supn∈N fn dμ < ∞ and condition (a) above holds (e.g., [],
page ). In the case of a finite measure, condition (b) trivially holds, and thus uniform
integrability implies equiintegrability. Conversely, provided that supn∈N fn dμ < ∞,
equiintegrability implies uniform integrability on each measurable set of finite measure; see [],
Proposition ., for related results.
3 A generalization of Fatou’s lemma
We are ready to state the main result of this paper.
Theorem . Let {fn}n∈N be a sequence of semiintegrable functions in L( ) such that
limn↑∞ fn is semiintegrable. Let {Bi}i∈N ⊂ F be a σ finite exhausting sequence. Suppose
that
Ai ⊂ Bi,
Then the Fatou inequality (.) holds.
Proof See Section .
It is shown in the proof (Lemma .) that (.) and (.) imply (.); i.e., (.) and (.)
imply that {Ai}i∈N is a σ finite exhausting sequence. Thus in Theorem ., the requirement
that {Ai} be a σ finite exhausting sequence can be replaced with (.). However, to verify
(.) to apply Theorem ., it is useful to have (.) instead of deriving it; for example, see
the proofs of Corollaries . and ..
If = Z+ and μ is the counting measure, we obtain a simple sufficient condition for the
Fatou inequality:
fn(t) ≤ ,
fn(t) ≤
where the sum is understood as the Lebesgue integral with respect to the counting
measure μ. Then
Proof Assume (.). For i ∈ N, let Bi = {, . . . , i – }. Then {Bi}i∈N is a σ finite exhausting
sequence. Let {Ai}i∈N ⊂ F satisfy (.). Then Ai = Bi for sufficiently large i. For such i we
have
fn(t) =
Hence (.) follows from (.). Now (.) holds by Theorem ..
Corollary . Suppose that = Z+ and that μ is the counting measure. Let {fn}n∈N be a
sequence of semiintegrable functions in L( ) such that limn↑∞ fn is semiintegrable. Suppose
further that
4 Known extensions of Fatou’s lemma
The version of Fatou’s lemma stated at the beginning of this paper can be shown as a
consequence of Theorem ..
Corollary . Let {fn}n∈N be a sequence in L( ) such that for some upper semiintegrable
function f ∈ L( ) we have fn ≤ f μa.e. for all n ∈ N. Then the Fatou inequality (.) holds.
Proof Since fn ≤ f μa.e. for all n ∈ N and f is upper semiintegrable, fn is upper
semiintegrable for each n ∈ N, and so is limn↑∞ fn. For any σ finite exhausting sequence {Ai}i∈N
we have
where the equality holds by (.) since f is upper semiintegrable. Now the Fatou inequality
(.) holds by Theorem ..
The following version of Fatou’s lemma is shown in [], page , and [], page , and can
be derived as a consequence of Theorem ..
Corollary . Suppose that μ( ) < ∞. Let {fn}n∈N be a sequence of functions in L( ) such
that {fn+}n∈N is uniformly integrable. Suppose further that limn↑∞ fn is semiintegrable. Then
the Fatou inequality (.) holds.
Proof Recall that uniform integrability of {fn+} requires integrability of each fn+ and
condition (a) in Section with fn+ replacing fn. Let {Ai}i∈N be any σ finite exhausting sequence.
where the equality holds by condition (a) since {fn+} is uniformly integrable and
limi↑∞ μ( \ Ai) = by (.) and the finiteness of μ. Now the Fatou inequality (.) holds
by Theorem ..
The next result is a slight variation on the results shown by [], Lemma . and [],
Corollary .. The latter results (unlike Corollary . below) do not require upper
semiintegrability of limn↑∞ fn since they use the upper integral, which always exists, instead of
the Lebesgue integral.
Corollary . Let {fn}n∈N be a sequence of integrable functions in L( ) such that {fn+}n∈N is
equiintegrable. Suppose that limn↑∞ fn is semiintegrable. Then the Fatou inequality (.)
holds.
Proof By equiintegrability of {fn+} and condition (b) in Section , there exists a sequence
{Ei}i∈N in F such that μ(Ei) < ∞ for all i ∈ N and
The first supremum on the righthand side converges to zero as i ↑ ∞ by (.) since Ei ⊂ Bi
for all i ∈ N. The second supremum also converges to zero as i ↑ ∞ by (.)(ii) and
condition (a) in Section . It follows that (.) holds for any sequence {Ai}i∈N in F satisfying
(.); thus by Theorem ., the Fatou inequality (.) holds.
5 Examples
In each of the examples below, is taken to be an interval in R. Accordingly, F is taken
to be the σ algebra of Lebesgue measurable subsets of , and μ the Lebesgue measure
restricted to F .
Our first example shows that Theorem . is a strict generalization of Corollaries .
and . even in the case of a finite measure.
Hence Corollary ., which requires uniform integrability of {fn+}, does not apply either.
Neither does Corollary . since equiintegrability implies uniform integrability on a finite
measure space provided that supn∈N fn dμ < ∞, which is the case here.
By contrast, Theorem . easily applies. To see this, note that, for each n ∈ N, fn is
integrable, and so is limn↑∞ fn. For i ∈ N, let
Bi = [–, –/i) ∪ (/i, ].
In the next example, μ is not finite, and the sequence {fn}n∈N is uniformly bounded from
below.
Example . Let
= R+. For n ∈ N, define fn :
Example . Let
= [–, ] \ {}. For n ∈ N, define fn :
It is easy to see that there is no upper semiintegrable function that dominates {fn}n∈N;
thus Corollary . does not apply. Furthermore, {fn+} is not uniformly integrable; indeed,
for any c ≥ we have
It is easy to see that there is no upper semiintegrable function that dominates {fn}n∈N;
thus Corollary . does not apply.
For any δ ∈ (, ) we have
n∈Z+
as n ↑ ∞.
Hence {fn+} does not satisfy condition (b) either. Therefore {fn+} is far from being
equiintegrable; as a consequence, Corollary . does not apply.
To see that Theorem . applies, note that, for each n ∈ N, fn is integrable for each n,
and so is limn↑∞ fn. For i ∈ N, let Bi = [, i). Then {Bi}i∈N is a σ finite exhausting sequence.
Take any sequence {Ai}i∈N in F satisfying (.)(i). Then for each fixed i ∈ N we have
\Ai fn dμ = for all n ≥ i. Thus the lefthand side of (.) equals zero. Hence the Fatou
inequality (.) holds by Theorem ..
In fact, as in the previous example, we have fn dμ = for all n ∈ N, and limn↑∞ fn = ;
thus both sides of the Fatou inequality (.) equal zero.
6 An application to infinitehorizon optimization in discrete time
In this section we consider a fairly general class of infinitehorizon maximization
problems, establishing a new result on the existence of an optimal path using Corollary .. We
start with some notation.
For t ∈ Z+, let Xt be a metric space. For t ∈ Z+, let t : Xt → Xt+ be a compactvalued
upper hemicontinuous correspondence in the sense that, for each x ∈ Xt , t(x) is a nonempty
compact subset of Xt+, and for any convergent sequence {xn}n∈N in Xt with limit x∗ ∈ Xt
and any sequence {yn}n∈N with yn ∈ t(xn) for all n ∈ N, there exists a convergent
subsequence {yni }i∈N of {yn}n∈N with limit y∗ ∈ t(x∗); see [], page and [], page ,
concerning this definition of upper hemicontinuity. For t ∈ Z+, let
Dt = (x, y) ∈ Xt × Xt+ : y ∈ t(x) .
For t ∈ Z+, let rt : Dt → R ∪ {–∞} be an upper semicontinuous function.
Consider the following maximization problem:
Thus {fn+} does not satisfy condition (a) in Section . To consider condition (b), let E ∈ F
with μ(E) < ∞. Then
s.t. xt+ ∈ t(xt),
x ∈ X given.
∀t ∈ Z+,
rt(xt, xt+) ≤
rt xt∗, xt∗+ ,
We say that a sequence {xt}t∞= is a feasible path (from x) if it satisfies (.). We say that a
feasible path {xt∗}t∞= is optimal (from x) if for any feasible path {xt}t∞=, we have
max rt(xt, xt+), < ∞.
We are ready to show our existence result.
Proposition . Let Assumption . hold. Suppose that, for any sequence {{xtn}t∞=}n∈N of
feasible paths, we have
rt xtn, xtn+ ≤ .
Then there exists an optimal path.
rt(xt, xt+),
where the supremum is taken over all feasible paths {xt}t∞=. By the definition of ν, there
exists a sequence {{xtn}t∞=}n∈N of feasible paths such that
nj
Since (x) is compact, there exists a convergent subsequence {x }j∈N of {xn}n∈N with
limit x∗ ∈ (x). By the definition of upper hemicontinuity, there exists a convergent
subsequence of {xnj }j∈N with limit x∗ ∈ (x∗). Continuing this way and using the
diagonal argument, we see that there exists a subsequence of {{xtn}t∞=}n∈N, again denoted by
{{xtn}t∞=}n∈N, such that, for each t ∈ N, xtn converges to some xt∗ as n ↑ ∞, and for each
t ∈ Z+, xt∗+ ∈ t(xt∗). Hence {xt∗}t∞= is a feasible path, which implies that
To apply Corollary ., let fn(t) = rt(xtn, xtn+) for t ∈ Z+. By Assumption ., for each n ∈ N,
fn(t) is an upper semiintegrable function of t ∈ Z+. For t ∈ Z+, let f ∗(t) = rt(xt∗, xt∗+). Since
{xt∗}t∞= is feasible as shown above, f ∗(t) is also an upper semiintegrable function of t ∈ Z+
by Assumption .. For each t ∈ Z+, by upper semicontinuity of rt we have
Since the rightmost side is an upper semiintegrable function of t ∈ Z+, so is the leftmost
side. Note that (.) directly follows from (.). Thus we can apply Corollary . to
obtain (.), which is written here as
rt xtn, xtn+ ≤
rt xt∗, xt∗+ ,
where (.) uses (.), and (.) uses (.). It follows from (.)(.) and (.) that
{xt∗}t∞= is an optimal path.
As a simple consequence of Proposition ., we obtain a result that can be viewed as an
abstract version of the existence result shown in [], Proposition .; see [], Theorem ,
for a similar result that requires stronger assumptions.
Corollary . Suppose that there exists an integrable function f : Z+ → R+ such that, for
any feasible path {xt}t∞=, we have
Then there exists an optimal path.
Proof Note that (.) implies Assumption .. Thus to apply Proposition ., it suffices
to verify (.) for an arbitrary sequence {{xtn}t∞=}n∈N of feasible paths. Let {{xtn}t∞=}n∈N be a
sequence of feasible paths. Then by (.) we have
f (t) = ,
where the last equality holds by integrability of f . It follows that (.) holds; hence an
optimal path exists by Proposition ..
Corollary . can be shown directly by using Fatou’s lemma to conclude (.) from
(.) in the proof of Proposition .. As illustrated in the next section, Proposition .
covers some important cases to which Corollary . fails to apply.
s.t. ct + xt+ = xt,
x ∈ R+ given.
ct, xt+ ≥ ,
∀t ∈ Z+,
t(x) = {y ∈ R+ : ≤ y ≤ x}.
It is easy to see from (.) that
∀t ∈ Z+, ct, xt ≤ x.
(i) ∀c ≥ ,
(ii) u(x) > .
In economics, u and δ are known as a utility function and a discount function, respectively.
The above maximization problem is a special case of (.)(.) such that, for all t ∈ Z+,
Xt = R+ and
7 Examples of optimization problems
To illustrate the significance of our existence result, we consider two special cases of the
following example.
Example . Let u : R+ → R ∪ {–∞} be a strictly increasing, upper semicontinuous
function. Let δ : R+ → R++ be a strictly decreasing function. Consider the following
maximization problem:
(Condition (ii) above does not depend on θ .) It is easy to see that condition (i) above implies
Assumption .; see (.)(.) for details.
Example . Consider Example .. Most discretetime economic models assume an
exponential discount function of the form
for some β ∈ (, ). In this case, Corollary . easily applies. To see this, let f (t) = βtu(x)
for t ∈ Z+. Then f : Z+ → R+ is integrable, and (.) holds by (.). Hence an optimal
path exists by Corollary ..
Example . Consider Example . again. Although exponential discounting (.) is
technically convenient (implying time consistency), experimental evidence suggests that
‘hyperbolic discounting’ is more plausible; see, e.g., [], page . A simple hyperbolic discount
function can be specified as follows:
In this example, Corollary . does not apply since there exists no integrable function
f : Z+ → R+ satisfying (.) for all feasible paths. To see this, define the feasible path
{x˜tn}t∞= for each n ∈ N by
⎧⎨ u(x)/( + αt) if t = n,
Hence any f satisfying (.) must satisfy
∀t ∈ Z+.
rt xtn, xtn+ =
t=i
θ
= .
where (.) uses (.)(i), and the second inequality in (.) uses (.). It follows that
Thus (.) holds; hence an optimal path exists by Proposition ..
In the above example, the hyperbolic discount function (.) is used to show that
Corollary . does not apply. The only property of the discount function required to apply
Proposition . is the equality in (.). We summarize this observation in the following
example.
Example . Consider Example . again. Suppose that
Then the argument of Example . shows that an optimal path exists by Proposition ..
8 Proof of Theorem 3.1
8.1 Preliminaries
Throughout the proof, we fix {fn}n∈N and {Bi}i∈N to be given by Theorem .. Define f ∗ =
limn↑∞ fn. For n ∈ N, define fˆn = supm≥n fm. We have
The following observation helps to simplify the proof.
Proof Suppose that f ∗ is not upper semiintegrable. Then (f ∗)+ dμ = ∞, and f ∗ must be
lower semiintegrable (i.e., (f ∗)– dμ < ∞) since f ∗ is semiintegrable by hypothesis. It
follows that f ∗ dμ = (f ∗)+ dμ – (f ∗)– dμ = ∞. Thus the Fatou inequality (.) trivially
holds.
Since the above result covers the case in which f ∗ is not upper semiintegrable, we
assume the following for the rest of the proof.
Assumption . f ∗ is upper semiintegrable.
8.2 Lemmas
We establish three lemmas before completing the proof of Theorem ..
Lemma . Suppose that there exists a σ finite exhausting sequence {Ai}i∈N satisfying (.)
and the following:
Then the Fatou inequality (.) holds.
Proof Since each fn is semiintegrable, we have
where (.) holds by (.), and (.) uses (.).
Since f ∗ is upper semiintegrable and {Aik }k∈N is a σ finite exhausting sequence, we have
limk↑∞ Aik f ∗ dμ = f ∗ dμ < ∞. Thus applying limk↑∞ to the righthand side of (.) yields
where the last inequality uses (.). The Fatou inequality (.) follows.
Lemma . Let {Ai}i∈N be a sequence in F such that, for each i ∈ N, μ(Ai) < ∞ and fˆn+
converges to (f ∗)+ uniformly on Ai as n ↑ ∞. Then {Ai}i∈N satisfies (.).
Proof Let i ∈ N. Let δ > . Since fˆn+ converges to (f ∗)+ uniformly on Ai as n ↑ ∞, for
sufficiently large n ∈ N we have fn ≤ fˆn ≤ fˆn+ ≤ (f ∗)+ + δ on Ai. Since (f ∗)+ is integrable by
Assumption . and μ(Ai) < ∞, (.) holds by Fatou’s lemma.
Lemma . Let {Ai}i∈N be a sequence in F satisfying (.) and (.). Then {Ai} is a σ finite
exhausting sequence.
Proof Since {Ai} satisfies (.) by hypothesis, it suffices to verify (.). For any i, j ∈ N with
i ≤ j, by (.) for {Bi}, we have
i∈N
i∈N
i∈N
i∈N
Ai = ,
μ(Bi \ Aj) ≤ μ(Bj \ Aj) → as j ↑ ∞,
where the convergence holds by (.). It follows that
j∈N
i∈N
j∈N
j∈N
Aj = .
i∈N Ai ⊂
i∈N Bi, we have
where the last equality holds by (.) for {Bi} and (.). It follows that {Ai}
satis
8.3 Completing the proof of Theorem 3.1
Ai =
Ej ⊂ Bi.
9 Conclusions
(.) also holds as shown in the previous paragraph, the Fatou inequality (.) holds by
In this paper we have provided a sufficient condition for what we call the Fatou inequality:
Our condition is considerably weaker than sufficient conditions known in the literature
such as uniform integrability (in the case of a finite measure) and equiintegrability. We
have illustrated the strength of our condition with simple examples. As an application, we
have shown a new result on the existence of an optimal path for deterministic
infinitehorizon optimization problems in discrete time. We have illustrated the strength of this
existence result with concrete examples of optimization problems.
Competing interests
The author declares that he has no competing interests.
Author’s contributions
This is a singleauthored paper. The author read and approved the final manuscript.
Acknowledgements
Financial support from the Japan Society for the Promotion of Science (JSPS KAKENHI Grant Number 15H05729) is
gratefully acknowledged.
1. Chow , YS, Robbins , H, Siegmund , D: Great Expectations : The Theory of Optimal Stopping . Houghton Mifflin , Boston ( 1971 )
2. Cairoli , R, Dalang, RC : Sequential Stochastic Optimization. Wiley, New York ( 1996 )
3. Denkowski , Z, Migórski , S, Papageorgiou, NS: An Introduction to Nonlinear Analysis: Theory . Springer, New York ( 2003 )
4. Shiryaev , AN: Optimal Stopping Rules . Springer, Berlin ( 2008 )
5. Giner , E: Calmness properties and contingent subgradients of integral functions on Lebesgue spaces Lp, 1 ≤ p < ∞ . SetValued Anal . 17 , 223  243 ( 2009 )
6. Giner , E: Fatou's lemma and lower epilimits of integral functions . J. Math. Anal. Appl . 394 , 13  29 ( 2012 )
7. Kamihigashi , T: Necessity of transversality conditions for infinite horizon problems . Econometrica 69 , 995  1012 ( 2001 )
8. Kamihigashi , T: Necessity of transversality conditions for stochastic problems . J. Econ. Theory 109 , 140  149 ( 2003 )
9. Kamihigashi , T: Necessity of transversality conditions for stochastic models with bounded or CRRA utility . J. Econ. Dyn. Control 29 , 1313  1329 ( 2005 )
10. Kamihigashi , T: Elementary results on solutions to the Bellman equation of dynamic programming: existence , uniqueness, and convergence. Econ. Theory 56 , 251  273 ( 2014 )
11. Cesari , L: OptimizationTheory and Applications . Springer, New York ( 1983 )
12. Ekeland , I, Scheinkman, JA : Transversality conditions for some infinite horizon discrete time optimization problems . Math. Oper. Res . 11 , 216  229 ( 1986 )
13. Brezis , H, Lieb , E: A relation between pointwise convergence of functions and convergence of functionals . Proc. Am. Math. Soc. 88 , 486  490 ( 1983 )
14. Balder , EJ: A unifying note on Fatou's lemma in several dimensions . Math. Oper. Res . 9 , 267  275 ( 1984 )
15. Loeb, PA, Sun , Y: A general Fatou lemma . Adv. Math. 213 , 741  762 ( 2007 )
16. Stokey , N, Lucas, RE Jr.: Recursive Methods in Economic Dynamics . Harvard University Press, Cambridge ( 1989 )
17. Aliprantis , CD, Border, KC: Infinite Dimensional Analysis: A Hitchhiker's Guide , 3rd edn. Springer, Berlin ( 2006 )
18. Le Van , C, Morhaim, L: Optimal growth models with bounded or unbounded returns: a unifying approach . J. Econ. Theory 105 , 158  187 ( 2002 )
19. Zarr , N, Alexander, WH, Brown, JW : Discounting of reward sequences: a test of competing formal models of hyperbolic discounting . Front. Psychol . 5 , 1  9 ( 2014 )