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.

Given a sequence [Formula: see text] of measurable functions on a σ-finite measure space such that the integral of each [Formula: see text] as well as that of [Formula: see text] exists in [Formula: see text], we provide a sufficient condition for the following inequality to hold: [Formula: see text] 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 infinite-horizon optimization problems in discrete time.

Introduction

Let be a measure space. Let be the set of measurable functions . A standard version of (reverse) Fatou’s lemma states that given a sequence in , if there exists an integrable function such that μ-a.e. for all , then where . 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 as well as that of exists, ‘uniform integrability’ of (where is the positive part of ) is a sufficient condition for the Fatou inequality (1.1) in the case of a finite measure (e.g., [1-4]); so is ‘equi-integrability’ of the same sequence in the case of a σ-finite measure (see [5, 6]). 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 based on the following assumption, which is maintained throughout the paper.

Assumption 1.1

is a σ-finite measure space.

Under this assumption there is an increasing sequence of measurable sets of 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 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 (1.1) as a corollary of our general result. Applying this condition to a fairly general class of infinite-horizon 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., [7-10]).

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., [11, 12]). 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., [13-15]). However, to our knowledge, there is no result in the literature that covers our generalization of 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.

Definitions

Given , let and denote the positive and negative parts of f, respectively; i.e., and . A function is called semi-integrable if or is integrable, and upper (lower) semi-integrable if () is integrable.

We say that a sequence in is a σ-finite exhausting sequence if 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 of integrable functions in is called equi-integrable (e.g., [6], page 16) if the following conditions hold:

For any there exists such that any with satisfies

For any there exists with such that

Suppose that . A sequence of integrable functions in is called uniformly integrable (e.g., [3], page 144) if It is well known that a sequence of integrable functions in is uniformly integrable if and only if and condition (a) above holds (e.g., [3], page 144). In the case of a finite measure, condition (b) trivially holds, and thus uniform integrability implies equi-integrability. Conversely, provided that , equi-integrability implies uniform integrability on each measurable set of finite measure; see [6], Proposition 2.8, for related results.

A generalization of Fatou’s lemma

We are ready to state the main result of this paper.

Theorem 3.1

Let be a sequence of semi-integrable functions in such that is semi-integrable. Let be a σ-finite exhausting sequence. Suppose that for any σ-finite exhausting sequence such that 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 is a σ-finite exhausting sequence. Thus in Theorem 3.1, the requirement that be a σ-finite exhausting sequence can 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 of Corollaries 4.1 and 4.2.

If and μ is the counting measure, we obtain a simple sufficient condition for the Fatou inequality:

Corollary 3.1

Suppose that and that μ is the counting measure. Let be a sequence of semi-integrable functions in such that is semi-integrable. Suppose further that where the sum is understood as the Lebesgue integral with respect to the counting measure μ. Then

Assume (3.3). For , let . Then is a σ-finite exhausting sequence. Let satisfy (3.2). Then for sufficiently large i. For such i we have Hence (3.1) follows from (3.3). Now (3.4) holds by Theorem 3.1. □

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 3.1.

Corollary 4.1

Let be a sequence in such that for some upper semi-integrable function we have μ-a.e. for all . Then the Fatou inequality (1.1) holds.

Since μ-a.e. for all and f is upper semi-integrable, is upper semi-integrable for each , and so is . For any σ-finite exhausting sequence we have where the equality holds by (2.2) since f is upper semi-integrable. Now the Fatou inequality (1.1) holds by Theorem 3.1. □

The following version of Fatou’s lemma is shown in [1], page 4, and [2], page 10, and can be derived as a consequence of Theorem 3.1.

Corollary 4.2

Suppose that . Let be a sequence of functions in such that is uniformly integrable. Suppose further that is semi-integrable. Then the Fatou inequality (1.1) holds.

Recall that uniform integrability of requires integrability of each and condition (a) in Section 2 with replacing . Let be any σ-finite exhausting sequence. We have where the equality holds by condition (a) since is uniformly integrable and 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 [5], Lemma 3.3 and [6], Corollary 3.3. The latter results (unlike Corollary 4.3 below) do not require upper semi-integrability of since they use the upper integral, which always exists, instead of the Lebesgue integral.

Corollary 4.3

Let be a sequence of integrable functions in such that is equi-integrable. Suppose that is semi-integrable. Then the Fatou inequality (1.1) holds.

By equi-integrability of and condition (b) in Section 2, there exists a sequence in such that for all and Since μ is σ-finite, there exists a σ-finite exhausting sequence . For , let . Then is also a σ-finite exhausting sequence. Let be a sequence in satisfying (3.2).

Fix for the moment. For each we have Applying to the leftmost and rightmost sides, we obtain The first supremum on the right-hand side converges to zero as by (4.3) since for all . The second supremum also converges to zero as by (3.2)(ii) and condition (a) in Section 2. It follows that (3.1) holds for any sequence in satisfying (3.2); thus by Theorem 3.1, the Fatou inequality (1.1) holds. □

Examples

In each of the examples below, Ω is taken to be an interval in . Accordingly, is taken to be the σ-algebra of Lebesgue measurable subsets of Ω, and μ the Lebesgue measure restricted to .

Our first example shows that Theorem 3.1 is a strict generalization of Corollaries 4.2 and 4.3 even in the case of a finite measure.

Example 5.1

Let . For , define by It is easy to see that there is no upper semi-integrable function that dominates ; thus Corollary 4.1 does not apply. Furthermore, is not uniformly integrable; indeed, for any we have Hence Corollary 4.2, which requires uniform integrability of , does not apply either. Neither does Corollary 4.3 since equi-integrability implies uniform integrability on a finite measure space provided that , which is the case here.

By contrast, Theorem 3.1 easily applies. To see this, note that, for each , is integrable, and so is . For , let Then is a σ-finite exhausting sequence. Let be any sequence in satisfying (3.2)(i). For each fixed , for any , we have on , and . Thus the left-hand side of (3.1) is zero. Hence the Fatou inequality (1.1) holds by Theorem 3.1.

In fact for all , and . Thus both sides of the Fatou inequality (1.1) equal zero.

In the next example, μ is not finite, and the sequence is uniformly bounded from below.

Example 5.2

Let . For , define by It is easy to see that there is no upper semi-integrable function that dominates ; thus Corollary 4.1 does not apply.

For any we have Thus does not satisfy condition (a) in Section 2. To consider condition (b), let with . Then which implies that . It follows that Hence does not satisfy condition (b) either. Therefore is far from being equi-integrable; as a consequence, Corollary 4.3 does not apply.

To see that Theorem 3.1 applies, note that, for each , is integrable for each n, and so is . For , let . Then is a σ-finite exhausting sequence. Take any sequence in satisfying (3.2)(i). Then for each fixed we have for all . Thus the left-hand side of (3.1) equals zero. Hence the Fatou inequality (1.1) holds by Theorem 3.1.

In fact, as in the previous example, we have for all , and ; thus both sides of the Fatou inequality (1.1) equal zero.

An application to infinite-horizon optimization in discrete time

In this section we consider a fairly general class of infinite-horizon maximization problems, establishing a new result on the existence of an optimal path using Corollary 3.1. We start with some notation.

For , let be a metric space. For , let be a compact-valued upper hemicontinuous correspondence in the sense that, for each , is a nonempty compact subset of , and for any convergent sequence in with limit and any sequence with for all , there exists a convergent subsequence of with limit ; see [16], page 56 and [17], page 564, concerning this definition of upper hemicontinuity. For , let For , let be an upper semicontinuous function.

Consider the following maximization problem: We say that a sequence is a feasible path (from ) if it satisfies (6.3). We say that a feasible path is optimal (from ) if for any feasible path , we have where . For the above inequality to make sense, we assume the following.

Assumption 6.1

For each feasible path , we have In other words, the mapping is upper semi-integrable.

We are ready to show our existence result.

Proposition 6.1

Let Assumption  6.1 hold. Suppose that, for any sequence of feasible paths, we have Then there exists an optimal path.

Let where the supremum is taken over all feasible paths . By the definition of ν, there exists a sequence of feasible paths such that Since is compact, there exists a convergent subsequence of with limit . By the definition of upper hemicontinuity, there exists a convergent subsequence of with limit . Continuing this way and using the diagonal argument, we see that there exists a subsequence of , again denoted by , such that, for each , converges to some as , and for each , . Hence is a feasible path, which implies that

To apply Corollary 3.1, let for . By Assumption 6.1, for each , is an upper semi-integrable function of . For , let . Since is feasible as shown above, is also an upper semi-integrable function of by Assumption 6.1. For each , by upper semicontinuity of we have Since the rightmost side is an upper semi-integrable function of , so is the leftmost side. Note that (3.3) directly follows from (6.7). Thus we can apply Corollary 3.1 to obtain (3.4), which is written here as

We are ready to show that is an optimal path. Recall from (6.9) that where (6.14) uses (6.12), and (6.15) uses (6.11). It follows from (6.13)-(6.15) and (6.10) that is an optimal path. □

As a simple consequence of Proposition 6.1, we obtain a result that can be viewed as an abstract version of the existence result shown in [12], Proposition 4.1; see [18], Theorem 1, for a similar result that requires stronger assumptions.

Corollary 6.1

Suppose that there exists an integrable function such that, for any feasible path , we have Then there exists an optimal path.

Note that (6.16) implies Assumption 6.1. Thus to apply Proposition 6.1, it suffices to verify (6.7) for an arbitrary sequence of feasible paths. Let be a sequence of feasible paths. Then by (6.16) we have where the last equality holds by integrability of f̅. 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.

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 be a strictly increasing, upper semicontinuous function. Let be a strictly decreasing function. Consider the following maximization problem: 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 , and It is easy to see from (7.2) that For simplicity, we assume that there exists such that (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 for some . In this case, Corollary 6.1 easily applies. To see this, let for . Then is integrable, and (6.16) holds by (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], page 1. A simple hyperbolic discount function can be specified as follows: for some .

In this example, Corollary 6.1 does not apply since there exists no integrable function satisfying (6.16) for all feasible paths. To see this, define the feasible path for each by Then Hence any f̅ satisfying (6.16) must satisfy Since the right-hand side is not upper semi-integrable in by (7.7)(ii), there exists no integrable function f̅ satisfying (6.16) for all feasible paths. Hence Corollary 6.1 does not apply.

However, Proposition 6.1 still applies. To see this, let be a sequence of feasible paths. For any we have where (7.14) uses (7.7)(i), and the second inequality in (7.16) uses (7.6). It follows that 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 to show that Corollary 6.1 does not apply. The only property of the discount function 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 that Then the argument of Example 7.3 shows that an optimal path exists by Proposition 6.1.

Proof of Theorem 3.1

Preliminaries

Throughout the proof, we fix and to be given by Theorem 3.1. Define . For , define . We have The following observation helps to simplify the proof.

Lemma 8.1

If is not upper semi-integrable, then the Fatou inequality (1.1) holds.

Proof

Suppose that is not upper semi-integrable. Then , and must be lower semi-integrable (i.e., ) since is semi-integrable by hypothesis. It follows that . Thus the Fatou inequality (1.1) trivially holds. □

Since the above result covers the case in which is not upper semi-integrable, we assume the following for the rest of the proof.

Assumption 8.1

is upper semi-integrable.

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 satisfying (3.1) and the following: Then the Fatou inequality (1.1) holds.

Since each is semi-integrable, we have By (3.1) there exists a subsequence of such that Note that is a σ-finite exhausting sequence.

Fix for the moment. Replacing i with in (8.3) and applying to both sides of the resulting equation, we obtain where (8.7) holds by (8.4), and (8.8) uses (8.2).

Since is upper semi-integrable and is a σ-finite exhausting sequence, we have . Thus applying to the right-hand side of (8.8) yields where the last inequality uses (8.5). The Fatou inequality (1.1) follows. □

Lemma 8.3

Let be a sequence in such that, for each , and converges to uniformly on as . Then satisfies (8.2).

Let . Let . Since converges to uniformly on as , for sufficiently large we have on . Since is integrable by Assumption 8.1 and , (8.2) holds by Fatou’s lemma. □

Lemma 8.4

Let be a sequence in satisfying (2.1) and (3.2). Then is a σ-finite exhausting sequence.

Since satisfies (2.1) by hypothesis, it suffices to verify (2.2). For any with , by (2.1) for , we have where the convergence holds by (3.2). It follows that Therefore Since , we have where the last equality holds by (2.2) for and (8.12). It follows that satisfies (2.2). □

Completing the proof of Theorem 3.1

Note from (8.1) that . Let be a sequence in such that . For each , by Egorov’s theorem there exists such that , , and converges to uniformly on as . For , let Then, for each , converges to uniformly on as . Thus (8.2) holds by Lemma 8.3.

Note that satisfies (2.1) and (3.2) by construction. Thus by Lemma 8.4, 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.

Conclusions

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 equi-integrability. 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 infinite-horizon optimization problems in discrete time. We have illustrated the strength of this existence result with concrete examples of optimization problems.