DP2016-37

A Generaliza tion of Fatou's Lemm a for
Extended Real-Valued Functions on
σ-Finite Measure Spaces: With an
Application to Infinite-Horizon
Optim iza tion in Discrete Time
Takashi KAMIHIGASHI
Revised January 10, 2017

A Generalization of Fatou’s Lemma for
Extended Real-Valued Functions on σ-Finite
Measure Spaces: With an Application to
Infinite-Horizon Optimization in Discrete Time
Takashi Kamihigashi∗
January 8, 2017

Abstract
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 oflim supn↑∞ fn exists in R, we provide a sufficient condition for thefollowing inequality to hold:Z
Z
lim sup fn dµ ≤ lim sup fn dµ.
n↑∞

n↑∞

Our condition is considerably weaker than sufficient conditions known
in the literature such as uniform integrability (in the case of a finite
measure) and equi-integrability. As an application, we obtain a new
result on the existence of an optimal path for deterministic infinitehorizon optimization problems in discrete time.
Keywords: Fatou’s lemma; σ-finite measure space; infinite-horizon
optimization; hyperbolic discounting; existence of optimal paths∗

RIEB, Kobe University, Rokkodai, Nada, Kobe 657-8501 JAPAN. Email:tkamihig@rieb.kobe-u.ac.jp. Tel/Fax: +81-78-803-7015.

1

Introduction

Let (Ω, F , µ) be a measure space. Let L(Ω) be the set of measurable functions f : Ω → R. A standard version of (reverse) Fatou’s lemma statesthat given a sequence {fn }n∈N in L(Ω), if there exists an integrable functionf ∈ L(Ω) such that fn ≤ f µ-a.e. for all n ∈ N, thenZ
Z
lim fn dµ ≤lim fn dµ,
(1.1)
n↑∞

n↑∞

where lim = lim sup. We call the above inequality the Fatou Inequality.
Some sufficient conditions for this inequality weaker than the one described above are known. In particular, provided that the integral of each fn
as well as that of limn↑∞ fn exists, “uniform integrability” of {fn+ } (where fn+
is the positive part of fn ) is a sufficient condition for the Fatou inequality (1.1)
in the case of a finite measure (e.g., [6, 4, 7, 17]); so is “equi-integrability”
of the same sequence in the case of a σ-finite measure (see [9, 10]). These
conditions are precisely defined in Section 2.
In this paper we provide a sufficient condition for the Fatou inequality(1.1) considerably weaker than the above conditions. Our approach is basedon the following assumption, which is maintained throughout the paper.
Assumption 1.1. (Ω, F , µ) is a σ-finite measure space.
Under this assumption there is an increasing sequence of measurable setsof finite measure whose union equals Ω. 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 measurespace is the set of nonnegative integers equipped with the counting measure.
In this setting, we provide a simple sufficient condition for the Fatou inequality (1.1) as a corollary of our general result. Applying this condition to afairly general class of infinite-horizon deterministic optimization problems indiscrete time, we establish a new result on the existence of an optimal path.
The condition takes a form similar to transversality conditions and otherrelated conditions in dynamic optimization (e.g., [11, 12, 13, 14]).
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., [5, 8]). 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.
1

We should mention that there are other important extensions of Fatou’s
lemma to more general functions and spaces (e.g., [3, 2, 16]). However, to ourknowledge, there is no result in the literature that covers our generalizationof Fatou’s lemma, which is specific to extended real-valued 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 equi-integrability. In Section 3 we state our main
result and derive those previous results as consequences. In Section 5 we
present two simple examples that cannot be treated by the previous results
but that can easily be treated using our result. In Section 6 we show a new
result on the existence of an optimal path for infinite-horizon deterministic optimization problems in discrete time. In Section 8 we prove our main
result.

2

Definitions

Given f ∈ L(Ω), let f + and f − denote the positive and negative partsof f , respectively; i.e., f + = max{f, 0} and f − = max{−f, 0}. A functionf ∈ L(Ω) is called semi-integrable if f + or f − is integrable, and upper (lower )semi-integrable if f + (f − ) is integrable.
We say that a sequence {Ai }i∈N in F is a σ-finite exhausting sequence if∀i ∈ N,Ai ⊂ Ai+1 , µ(Ai ) < ∞,µ(Ω \ ∪i∈N Ai ) = 0.

(2.1)
(2.2)

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(Ω) is called equi-integrable(e.g., [10, p. 16]) if the following conditions hold:(a) For any  > 0 there exists δ > 0 such that any A ∈ F with µ(A) < δ
satisfiesZ
sup |fn | dµ ≤ .
(2.3)
n∈N

A

(b) For any  > 0 there exists E ∈ F with µ(E) < ∞ such thatZ
sup|fn | dµ ≤ .
n∈N

Ω\E

2

(2.4)

Suppose that µ(Ω) < ∞. A sequence {fn }n∈N of integrable functions in
L(Ω) is called uniformly integrable (e.g., [7, p. 144]) ifZlim sup|fn | dµ = 0.
(2.5)
c↑∞

n∈N

{|fn |≥c}

It is known that a sequence {fn }n∈N of
R integrable functions in L(Ω) is uniformly integrable if and only if supn∈N |fn | dµ < ∞ and condition (a) above
holds (e.g., [7, p. 144]). In the case of a finite measure, condition (b) triviallyholds, and thus uniform
R integrability implies equi-integrability. Conversely,
provided that supn∈N |fn | dµ < ∞, equi-integrability implies uniform integrability on each measurable set of finite measure; see [10, Proposition 2.8]
for related results.

3

A Generalization of Fatou’s Lemma

We are ready to state the main result of this paper.
Theorem 3.1. Let {fn }n∈N be a sequence of semi-integrable functions in
L(Ω) such that limn↑∞ fn is semi-integrable. Let {Bi }i∈N ⊂ F be a σ-finite
exhausting sequence. Suppose thatZ
fn dµ ≤ 0
(3.1)
lim limi↑∞ n↑∞Ω\Aifor any σ-finite exhausting sequence {Ai }i∈N ⊂ F such that(i) ∀i ∈ N, Ai ⊂ Bi ,(ii) lim µ(Bi \ Ai ) = 0.
i↑∞

(3.2)

Then the Fatou inequality (1.1) holds.
Proof. See Section 8.
It is shown in the proof (Lemma 8.4) that (2.1) and (3.2) imply (2.2); i.e.,(2.1) and (3.2) imply that {Ai }i∈N is a σ-finite exhausting sequence. Thus
in Theorem 3.1, the requirement that {Ai } be a σ-finite exhausting sequencecan be replaced with (2.1). However, to verify (3.1) to apply Theorem 3.1,
it is useful to have (2.2) instead of deriving it; for example, see the proofs ofCorollaries 4.1 and 4.2.
If Ω = Z+ and µ is the counting measure, we obtain a simple sufficientcondition for the Fatou inequality:3

Corollary 3.1. Suppose that Ω = Z+ and that µ is the counting measure.
Let {fn }n∈N be a sequence of semi-integrable functions in L(Ω) such thatlimn↑∞ fn is semi-integrable. Suppose further that
lim lim∞Xi↑∞ n↑∞ t=ifn (t) ≤ 0,

(3.3)

where the sum is understood as the Lebesgue integral with respect to the counting measure µ. Then
lim

n↑∞

∞
X

fn (t) ≤

t=0

∞
X
t=0

lim fn (t).

n↑∞

(3.4)

Proof. Assume (3.3). For i ∈ N, let Bi = {0, . . . , i − 1}. Then {Bi }i∈N is a
σ-finite exhausting sequence. Let {Ai }i∈N ⊂ F satisfy (3.2). Then Ai = Bi
for sufficiently large i. For such i we haveX

fn (t) =∞X

fn (t).

(3.5)

t=i

Ω\Ai

Hence (3.1) follows from (3.3). Now (3.4) holds by Theorem 3.1.

4Known Extensions of Fatou’s Lemma

The version of Fatou’s lemma stated at the beginning of this paper can beshown as a consequence of Theorem 3.1.
Corollary 4.1. Let {fn }n∈N be a sequence in L(Ω) such that for some uppersemi-integrable function f ∈ L(Ω) we have fn ≤ f µ-a.e. for all n ∈ N. Thenthe Fatou inequality (1.1) holds.
Proof. Since fn ≤ f µ-a.e. for all n ∈ N and f is upper semi-integrable, fn isupper semi-integrable for each n ∈ N, and so is limn↑∞ fn . For any σ-finite
exhausting sequence {Ai }i∈N we haveZ
Z
Z
lim limfn dµ ≤ lim
f dµ ≤ lim
f + dµ = 0,
(4.1)
i↑∞ n↑∞

Ω\Ai

i↑∞

i↑∞

Ω\Ai

Ω\Ai

where the equality holds by (2.2) since f is upper semi-integrable. Now theFatou inequality (1.1) holds by Theorem 3.1.
4

The following version of Fatou’s lemma is shown in [6, p. 4] and [4, p.
10], and can be derived as a consequence of Theorem 3.1.
Corollary 4.2. Suppose that µ(Ω) < ∞. Let {fn }n∈N be a sequence offunctions in L(Ω) such that {fn+ }n∈N is uniformly integrable. Suppose further
that limn↑∞ fn is semi-integrable. Then the Fatou inequality (1.1) holds.
Proof. Recall that uniform integrability of {fn+ } requires integrability of eachfn+ and condition (a) in Section 2 with fn+ replacing fn . Let {Ai }i∈N be any
σ-finite exhausting sequence. We haveZ
Z
lim limfn dµ ≤ lim sup
fn+ dµ = 0,
(4.2)
i↑∞ n∈NΩ\Aii↑∞ n∈NΩ\Aiwhere the equality holds by condition (a) since {fn+ } is uniformly integrable
and limi↑∞ µ(Ω \ Ai ) = 0 by (2.2) and the finiteness of µ. Now the Fatou
inequality (1.1) holds by Theorem 3.1.
The next result is a slight variation on the results shown by [9, Lemma
3.3] and [10, Corollary 3.3]. These latter results (unlike Corollary 4.3 below)
do not require upper semi-integrability of limn↑∞ fn since they use the upper
integral, which always exists, instead of the Lebesgue integral.
Corollary 4.3. Let {fn }n∈N be a sequence of integrable functions in L(Ω)
such that {fn+ }n∈N is equi-integrable. Suppose that limn↑∞ fn is semi-integrable.
Then the Fatou inequality (1.1) holds.
Proof. By equi-integrability of {fn+ } and condition (b) in Section 2, there
exists a sequence {Ei }i∈N in F such that µ(Ei ) < ∞ for all i ∈ N andZ
lim supfn+ dµ = 0.
(4.3)
i↑∞ n∈NΩ\EiSince µ is σ-finite, there exists a σ-finite exhausting sequence {Ci }i∈N . Fori ∈ N, let Bi = (∪ij=1 Ej ) ∪ Ci . Then {Bi }i∈N is also a σ-finite exhausting
sequence. Let {Ai }i∈N be a sequence in F satisfying (3.2).
Fix i ∈ N for the moment. For each n ∈ N we have
Z
Z
Z
Z
+
+
fn dµ ≤
fn dµ =
fn dµ +
fn+ dµ.
(4.4)
Ω\Ai

Ω\Ai

Ω\Bi

5

Bi \Ai

Applying supn∈N to the leftmost and rightmost sides, we obtain
Z
Z
Z
+
sup
fn ≤ supfn dµ + sup
fn+ dµ.
n∈N

Ω\Ai

n∈N

n∈N

Ω\Bi

(4.5)

Bi \Ai

The first supremum on the right-hand side converges to zero as i ↑ ∞ by(4.3) since Ei ⊂ Bi for all i ∈ N. The second supremum also converges tozero as i ↑ ∞ by (3.2)(ii) and condition (a) in Section 2. It follows that (3.1)
holds for any sequence {Ai }i∈N in F satisfying (3.2); thus by Theorem 3.1,
the Fatou inequality (1.1) holds.

5

ExamplesIn each of the examples below, Ω is taken to be an interval in R. Accordingly,
F is taken to be the σ-algebra of Lebesgue measurable subsets of Ω, and µ
the Lebesgue measure restricted to F .
Our first example shows that Theorem 3.1 is a strict generalization ofCorollaries 4.2 and 4.3 even in the case of a finite measure.
Example 5.1. Let Ω = [−1, 1] \ {0}. For n ∈ N, define fn : Ω → R by

0



−n
fn (ω) =

n



0

if ω ∈ [−1, −1/n),
if ω ∈ [−1/n, 0),
if ω ∈ (0, 1/n],
if ω ∈ (1/n, 1].

(5.1)

It is easy to see that there is no upper semi-integrable function that dominates {fn }n∈N ; thus Corollary 4.1 does not apply. Furthermore, {fn+ } is not
uniformly integrable; indeed, for any c ≥ 0 we haveZsupfn+ dµ = sup n/n = 1.
(5.2)
n∈N

{fn+ ≥c}n∈N:n≥c

Hence Corollary 4.2, which requires uniform integrability of {fn+ }, does not
apply either. Neither does Corollary 4.3 since equi-integrability implies
uniR
form integrability on a finite measure space provided that supn∈N |fn | dµ <∞, which is the case here.

6By contrast, Theorem 3.1 easily applies. To see this, note that for eachn ∈ N, fn is integrable, and so is limn↑∞ fn . For i ∈ N, let
Bi = [−1, −1/i) ∪ (1/i, 1].

(5.3)

Then {Bi }i∈N is a σ-finite exhausting sequence. Let {Ai }i∈N be any sequencein F satisfying
R (3.2)(i). For
R each fixed i ∈ N, for any n ≥ i, we have fn = 0
on Bi , and Ω\Ai fn dµ = Ω\Bi fn dµ = 0. Thus the left-hand side of (3.1) iszero. HenceR the Fatou inequality (1.1) holds by Theorem 3.1.
In fact fn dµ = 0 for all n ∈ N, and limn↑∞ fn = 0. Thus both sides of
the Fatou inequality (1.1) equal zero.
In the next example, µ is not finite, and the sequence {fn }n∈N is uniformly
bounded from below.
Example 5.2. Let Ω = R+ . For n ∈ N, define fn : Ω → R by

0



n
fn (ω) =

−1



0

if ω ∈ [0, n),if ω ∈ [n, n + 1),
if ω ∈ [n + 1, 2n + 1),
if ω ≥ 2n + 1.

(5.4)

It is easy to see that there is no upper semi-integrable function that dominates{fn }n∈N ; thus Corollary 4.1 does not apply.
For any δ ∈ (0, 1) we haveZ
fn+ dµ = nδ ↑ ∞ as n ↑ ∞.
(5.5)
[n,n+δ)Thus {fn+ } does not satisfy condition (a) in Section 2. To consider condition(b), let E ∈ F with µ(E) < ∞. ThenXµ(E) =µ(E ∩ [n, n + 1)) < ∞,
(5.6)
n∈Z+

which implies that limn↑∞ µ(E ∩ [n, n + 1)) = 0. It follows thatZ
fn+ dµ = n(1 − µ(E ∩ [n, n + 1))) → ∞ as n ↑ ∞.
Ω\E

7

(5.7)

Hence {fn+ } does not satisfy condition (b) either. Therefore {fn+ } is far frombeing equi-integrable; as a consequence, Corollary 4.3 does not apply.
To see that Theorem 3.1 applies, note that for each n ∈ N, fn is integrable
for each n, and so is limn↑∞ fn . For i ∈ N, let Bi = [0, i). Then {Bi }i∈N is
a σ-finite exhausting sequence. Take any sequence{Ai }i∈N in F satisfyingR(3.2)(i). Then for each fixed i ∈ N we have Ω\Ai fn dµ = 0 for all n ≥ i.
Thus the left-hand side of (3.1) equals zero. Hence the Fatou inequality (1.1)
holds by Theorem 3.1.
R
In fact, as in the previous example, we have fn dµ = 0 for all n ∈ N,
and limn↑∞ fn = 0; thus both sides of the Fatou inequality (1.1) equal zero.

6

An Application to Infinite-Horizon Optimization in Discrete TimeIn this section we consider a fairly general class of infinite-horizon maximization problems, establishing a new result on the existence of an optimal pathusing Corollary 3.1. We start with some notation.
For t ∈ Z+ , let Xt be a metric space. For t ∈ Z+ , let Γt : Xt → Xt+1
be a compact-valued upper hemicontinuous correspondence in the sense thatfor each x ∈ Xt , Γt (x) is a nonempty compact subset of Xt+1 , and for any
convergent sequence {xn }n∈N in Xt with limit x∗ ∈ Xt and any sequence{yn }n∈N with yn ∈ Γt (xn ) for all n ∈ N, there exists a convergent subsequence{yni }i∈N of {yn }n∈N with limit y ∗ ∈ Γt (x∗ ); see [18, p. 56] and [1, p. 564]
concerning this definition of upper hemicontinuity. For t ∈ Z+ , let
Dt = {(x, y) ∈ Xt × Xt+1 : y ∈ Γt (x)}.

(6.1)

For t ∈ Z+ , let rt : Dt → R ∪ {−∞} be an upper semicontinuous function.
Consider the following maximization problem:
max

∞
X

rt (xt , xt+1 )(6.2)

s.t. xt+1 ∈ Γt (xt ), ∀t ∈ Z+ ,x0 ∈ X0 given.

(6.3)
(6.4)

{xt }∞
t=1

t=0

We say that a sequence {xt }∞t=1 is a feasible path (from x0 ) if it satisfies (6.3).
We say that a feasible path {x∗t }∞t=1 is optimal (from x0 ) if for any feasible
8path {xt }∞
t=1 , we have∞Xrt (xt , xt+1 ) ≤

t=0

∞
X

rt (x∗t , x∗t+1 ),

(6.5)

t=0

where x∗0 = x0 . For the above inequality to make sense, we assume thefollowing.
Assumption 6.1. For each feasible path {xt }∞
t=1 , we have∞X

max{rt (xt , xt+1 ), 0} < ∞.

(6.6)

t=0

In other words, the mapping rt (xt , xt+1 ) : t 7→ R ∪ {−∞} is upper semiintegrable.
We are ready to show our existence result.
Proposition 6.1. Let Assumption 6.1 hold. Suppose that for any sequence{{xnt }∞
t=1 }n∈N of feasible paths, we have
lim limi↑∞ n↑∞∞Xrt (xnt , xnt+1 ) ≤ 0.

(6.7)

t=i

Then there exists an optimal path.
Proof. Letν = sup∞Xrt (xt , xt+1 ),

(6.8)

t=0

where the supremum is taken over all feasible paths {xt }∞t=1 . By the definitionof ν, there exists a sequence {{xnt }∞}offeasiblepaths
such thatt=1 n∈N
lim

n↑∞

∞
X

rt (xnt , xnt+1 ) = ν.

(6.9)

t=0
n

Since Γ0 (x0 ) is compact, there exists a convergent subsequence {x1 j }j∈N of{xn1 }n∈N with limit x∗1 ∈ Γ0 (x0 ). By the definition of upper hemicontinuity,nthere exists a convergent subsequence of {x2 j }j∈N with limit x∗2 ∈ Γ1 (x∗1 ).
9

Continuing this way and using the diagonal argument, we see that theren ∞exists a subsequence of {{xnt }∞t=1 }n∈N , again denoted by {{xt }t=1 }n∈N , suchthat for each t ∈ N, xnt converges to some x∗t as n ↑ ∞, and for each t ∈ Z+ ,x∗t+1 ∈ Γt (x∗t ). Hence {x∗t }∞t=1 is a feasible path, which implies that∞X

rt (x∗t , x∗t+1 ) ≤ ν.

(6.10)

t=0

To apply Corollary 3.1, let fn (t) = rt (xnt , xnt+1 ) for t ∈ Z+ . By Assumption
6.1, for each n ∈ N, fn (t) is an upper semi-integrable function of t ∈ Z+ .
For t ∈ Z+ , let f ∗ (t) = rt (x∗t , x∗t+1 ). Since {x∗t }∞t=1 is feasible as shown above,f ∗ (t) is also an upper semi-integrable function of t ∈ Z+ by Assumption 6.1.
For each t ∈ Z+ , by upper semicontinuity of rt we have
lim fn (t) = lim rt (xnt , xnt+1 ) ≤ rt (x∗t , x∗t+1 ) = f ∗ (t).

n↑∞

n↑∞

(6.11)

Since the rightmost side is an upper semi-integrable function of t ∈ Z+ , so is
the leftmost side. Note that (3.3) directly follows from (6.7). Thus we canapply Corollary 3.1 to obtain (3.4), which is written here as
lim

n↑∞

∞
X

rt (xnt , xnt+1 ) ≤

∞
X
t=0

t=0

lim rt (xnt , xnt+1 ).

n↑∞(6.12)We are ready to show that {x∗t }∞t=1 is an optimal path. Recall from (6.9)
that∞Xν = limrt (xnt , xnt+1 )(6.13)
n↑∞

≤
≤

∞
X
t=0
∞
X

t=0

lim rt (xnt , xnt+1 )(6.14)rt (x∗t , x∗t+1 ),(6.15)

n↑∞t=0

where (6.14) uses (6.12), and (6.15) uses (6.11). It follows from (6.13)–(6.15)
and (6.10) that {x∗t }∞t=1 is an optimal path.
As a simple consequence of Proposition 6.1, we obtain a result that canbe viewed as an abstract version of the existence result shown in [8, Proposition 4.1]; see [15, Theorem 1] for a similar result that requires stronger
assumptions.
10Corollary 6.1. Suppose that there exists an integrable function f : Z+ → R+such that for any feasible path {xt }∞
t=1 , we have∀t ∈ Zt ,rt (xt , xt+1 ) ≤ f (t).

(6.16)

Then there exists an optimal path.
Proof. Note that (6.16) implies Assumption 6.1. Thus to apply Proposition
6.1, it suffices to verify (6.7) for an arbitrary sequence {{xnt }∞t=1 }n∈N of feasiblen ∞paths. Let {{xt }t=1 }n∈N be a sequence of feasible paths. Then by (6.16) wehave
lim lim∞Xi↑∞ n↑∞ t=irt (xnt , xnt+1 ) ≤ lim limi↑∞ n↑∞∞
X
t=i

f (t) = lim∞Xf (t) = 0,

(6.17)i↑∞ t=iwhere the last equality holds by integrability of f . It follows that (6.7) holds;
hence an optimal path exists by Proposition 6.1.
Corollary 6.1 can be shown directly by using Fatou’s lemma to conclude(6.12) from (6.16) in the proof of Proposition 6.1. As illustrated in the next
section, Proposition 6.1 covers some important cases to which Corollary 6.1
fails to apply.

7

Examples of Optimization Problems

To illustrate the significance of our existence result, we consider two special
cases of the following example.
Example 7.1. Let u : R+ → R ∪ {−∞} be a strictly increasing, uppersemicontinuous function. Let δ : R+ → R++ be a strictly decreasing function.
Consider the following maximization problem:max{xt }∞
t=1∞
X

δ(t)u(ct )

(7.1)

t=0

s.t. ct + xt+1 = xt , ct , xt+1 ≥ 0,x0 ∈ R+ given.

11∀t ∈ Z+ ,

(7.2)
(7.3)

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 (6.2)–(6.4)
such that for all t ∈ Z+ , Xt = R+ andrt (x, y) = δ(t)u(x − y),Γt (x) = {y ∈ R+ : 0 ≤ y ≤ x}.

(7.4)
(7.5)

It is easy to see from (7.2) that∀t ∈ Z+ ,ct , xt ≤ x0 .

(7.6)

For simplicity, we assume that there exists θ > 0 such that(i) ∀c ≥ 0, u(c) ≤ θc,(ii) u(x0 ) > 0.

(7.7)

(Condition (ii) above does not depend on θ.) It is easy to see that condition(i) above implies Assumption 6.1; see (7.13)–(7.16) for details.
Example 7.2. Consider Example 7.1. Most discrete-time economic models
assume an exponential discount function of the form
δ(t) = β t∀t ∈ Z+ ,(7.8)

for some β ∈ (0, 1). In this case, Corollary 6.1 easily applies. To see this, let
f (t) = β t u(x0 ) for t ∈ Z+ . Then f : Z+ → R+ is integrable, and (6.16) holdsby (7.6). Hence an optimal path exists by Corollary 6.1.
Example 7.3. Consider Example 7.1 again. Although exponential discounting (7.8) is technically convenient (implying time consistency), experimental
evidence suggests that “hyperbolic discounting” is more plausible; see, e.g.,[19, p. 1]. A simple hyperbolic discount function can be specified as follows:∀t ∈ Z+ ,δ(t) =

1
1 + αt(7.9)

for some α > 0.
In this example, Corollary 6.1 does not apply since there exists no integrable function f : Z+ → R+ satisfying (6.16) for all feasible paths. To see
this, define the feasible path {x̃nt }∞t=1 for each n ∈ N by(
x0 if t ≤ n,x̃nt =(7.10)
0 if t ≥ n + 1.
12

Then
(
u(x0 )/(1 + αt) if t = n,
rt (x̃nt , x̃nt+1 ) =u(0)/(1 + αt) otherwise.

(7.11)

Hence any f satisfying (6.16) must satisfy
f (t) ≥ u(x0 )/(1 + αt),∀t ∈ Z+ .

(7.12)

Since the right-hand side is not upper semi-integrable in t ∈ Z+ by (7.7)(ii),there exists no integrable function f satisfying (6.16) for all feasible paths.
Hence Corollary 6.1 does not apply.
However, Proposition 6.1 still applies. To see this, let {{xnt }∞t=1 }n∈N be asequence of feasible paths. For any n, i ∈ N we have∞∞XXu(xnt − xnt+1 )nn(7.13)rt (xt , xt+1 ) =1 + αt
t=i
t=i
≤

∞
X
θ(xnt − xnt+1 )1 + αit=i

θ(7.14)

P∞nnt=i (xt − xt+1 )(7.15)
1 + αi
θxni
θx0
≤
≤
,
(7.16)
1 + αi1 + αiwhere (7.14) uses (7.7)(i), and the second inequality in (7.16) uses (7.6). It
follows that
∞
X
θx0
= 0.
(7.17)
lim limrt (xnt , xnt+1 ) ≤ limi↑∞ n↑∞ t=ii↑∞ 1 + αi=

Thus (6.7) holds; hence an optimal path exists by Proposition 6.1.
In the above example, the hyperbolic discount function (7.9) is used toshow that Corollary 6.1 does not apply. The only property of the discountfunction required to apply Proposition 6.1 is the equality in (7.17). We
summarize this observation in the following example.
Example 7.4. Consider Example 7.1 again. Suppose thatlim δ(t) = 0.
t↑∞

(7.18)

Then the argument of Example 7.3 shows that an optimal path exists by
Proposition 6.1.
13

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
3.1. Define f ∗ = limn↑∞ fn . For n ∈ N, define fˆn = supm≥n fm . We havef ∗ = lim fˆn .

(8.1)

n↑∞

The following observation helps simplify the proof.
Lemma 8.1. If f ∗ is not upper semi-integrable, then the Fatou inequality(1.1) holds.
R ∗ +Proof. Suppose that f ∗ is not upper semi-integrable.
Then(f ) dµ = ∞,R ∗ −∗and f must be lower semi-integrable (i.e.,sincef ∗ is semiR ∗(f ) dµR <∗ ∞)
R
integrable by hypothesis. It follows that f dµ = (f )+ dµ − (f ∗ )− dµ =∞. Thus the Fatou inequality (1.1) 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 8.1. f ∗ is upper semi-integrable.

8.2

Lemmas

We establish three lemmas before completing the proof of Theorem 3.1.
Lemma 8.2. Suppose that there exists a σ-finite exhausting sequence {Ai }i∈Nsatisfying (3.1) and the following:Z
Z∀i ∈ N, lim
fn dµ ≤f ∗ dµ.
(8.2)
n↑∞

Ai

Ai

Then the Fatou inequality (1.1) holds.
Proof. Since each fn is semi-integrable, we haveZZ
Z∀i, n ∈ N,
fn dµ =
fn dµ +
Ai

14

Ω\Ai

fn dµ.

(8.3)

By (3.1) there exists a subsequence {Aik }k∈N of {Ai }i∈N such thatZ∀k ∈ N, lim
fn dµ < ∞,
n↑∞

(8.4)

Ω\Aik

Z
fn dµ ≤ 0.

lim limk↑∞ n↑∞

(8.5)

Ω\Aik

Note that {Aik }k∈N is a σ-finite exhausting sequence.
Fix k ∈ N for the moment. Replacing i with ik in (8.3) and applying
limn↑∞ to both sides of the resulting equation, we obtain
#
"Z
Z
Z
lim

n↑∞

fn dµ

fn dµ +

fn dµ = lim

n↑∞

Z

Z

≤ lim

n↑∞

Z
≤

(8.6)

Ω\Aik

Aik

fn dµ + lim

n↑∞

Aik

fn dµ

(8.7)

Ω\Aik

Z

∗

f dµ + lim

n↑∞

Aik

fn dµ,

(8.8)

Ω\Aik

where (8.7) holds by (8.4), and (8.8) uses (8.2).
Since f ∗ is upper semi-integrable
andR ∗{Aik }k∈N is a σ-finite exhaustingR∗
sequence, we have limk↑∞ Ai f dµ = f dµ < ∞. Thus applying limk↑∞kto the right-hand side of (8.8) yields
Z
Z
Z
Z
∗
fn dµ ≤ f ∗ dµ,(8.9)
lim fn dµ ≤ f dµ + lim limn↑∞k↑∞ n↑∞Ω\Aikwhere the last inequality uses (8.5). The Fatou inequality (1.1) follows.
Lemma 8.3. 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 (8.2).
Proof. Let i ∈ N. Let δ > 0. 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 8.1 and µ(Ai ) < ∞, (8.2) holds
by Fatou’s lemma.
Lemma 8.4. Let {Ai }i∈N be a sequence in F satisfying (2.1) and (3.2).
Then {Ai } is a σ-finite exhausting sequence.
15

Proof. Since {Ai } satisfies (2.1) by hypothesis, it suffices to verify (2.2). For
any i, j ∈ N with i ≤ j, by (2.1) for {Bi }, we haveµ(Bi \ Aj ) ≤ µ(Bj \ Aj ) → 0 as j ↑ ∞,(8.10)

where the convergence holds by (3.2). It follows that∀i ∈ N,µ(Bi \ ∪j∈N Aj ) = lim µ(Bi \ Aj ) = 0.

(8.11)µ(∪i∈N Bi \ ∪j∈N Aj ) = lim µ(Bi \ ∪j∈N Aj ) = 0.

(8.12)j↑∞

Thereforei↑∞Since ∪i∈N Ai ⊂ ∪i∈N Bi , we haveµ(Ω \ ∪i∈N Ai ) = µ(Ω \ ∪i∈N Bi ) + µ(∪i∈N Bi \ ∪i∈N Ai ) = 0,(8.13)

where the last equality holds by (2.2) for {Bi } and (8.12). It follows that{Ai } satisfies (2.2).

8.3

Completing the Proof of Theorem 3.1

Note from (8.1) that (f ∗ )+ = limn↑∞ fˆn+ . Let {i }i∈N be a sequence in R++
such that limi↑∞ i = 0. For each i ∈ N, by Egorov’s theorem there exists
Ei ∈ F such that Ei ⊂ Bi , µ(Bi \ Ei ) < i , and fˆn+ converges to (f ∗ )+uniformly on Ei as n ↑ ∞. For i ∈ N, let
Ai = ∪ij=1 Ej ⊂ Bi .

(8.14)

Then for each i ∈ N, fˆn+ converges to (f ∗ )+ uniformly on Ai as n ↑ ∞. Thus
(8.2) holds by Lemma 8.3.
Note that {Ai }i∈N satisfies (2.1) and (3.2) by construction. Thus by
Lemma 8.4, {Ai } is a σ-finite exhausting sequence. Hence (3.1) holds by the
hypothesis of Theorem 3.1. Since (8.2) also holds as shown in the previous
paragraph, the Fatou inequality (1.1) holds by Lemma 8.2.

16

9Conclusions

In this paper we have provided a sufficient condition for what we call the
Fatou inequality:
Z
Z
lim

n↑∞

fn dµ ≤

lim fn dµ.

n↑∞

Our condition is considerably weaker than sufficient conditions known in theliterature such as uniform integrability (in the case of a finite measure) andequi-integrability. We have illustrated the strength of our condition withsimple examples. As an application, we have shown a new result on theexistence of an optimal path for deterministic infinite-horizon optimization
problems in discrete time. We have illustrated the strength of this existence
result with concrete examples of optimization problems.

10

Competing Interests

The author declares that he has no competing interests.

11

Acknowledgements

Financial support from the Japan Society for the Promotion of Science (JSPS
KAKENHI Grant Number 15H05729) is gratefully acknowledged.

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