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

Journal of Inequalities and Applications, Jan 2017

Given a sequence { f n } n ∈ N of measurable functions on a σ-finite measure space such that the integral of each f n as well as that of lim sup n ↑ ∞ f n exists in R ‾ , we provide a sufficient condition for the following inequality to hold: lim sup n ↑ ∞ ∫ f n d μ ≤ ∫ lim sup n ↑ ∞ f n d μ . 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.

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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

Kamihigashi Journal of Inequalities and Applications 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 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; infinite-horizon 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 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. 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 ‘equi-integrability’ 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 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., [–]). 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 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 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 infinite-horizon 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 equi-integrable (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 equi-integrability. 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 semi-integrable functions in L( ) such that limn↑∞ fn is semi-integrable. 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 semi-integrable functions in L( ) such that limn↑∞ fn is semi-integrable. 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 semi-integrable function f ∈ L( ) we have fn ≤ f μ-a.e. for all n ∈ N. Then the Fatou inequality (.) holds. Proof Since fn ≤ f μ-a.e. for all n ∈ N and f is upper semi-integrable, 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 semi-integrable. 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 semi-integrable. 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 equi-integrable. Suppose that limn↑∞ fn is semi-integrable. Then the Fatou inequality (.) holds. Proof By equi-integrability 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 right-hand 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 equi-integrability 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 semi-integrable 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 semi-integrable 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 left-hand 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 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 .. We start with some notation. For t ∈ Z+, let Xt be a metric space. For t ∈ Z+, let t : Xt → Xt+ be a compact-valued 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 {xn}n∈N with limit x∗ ∈ (x). By the definition of upper hemicontinuity, there exists a convergent subsequence of {xnj }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 semi-integrable 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 semi-integrable function of t ∈ Z+ by Assumption .. For each t ∈ Z+, by upper semicontinuity of rt we have Since the rightmost side is an upper semi-integrable 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 discrete-time 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 semi-integrable. Then (f ∗)+ dμ = ∞, and f ∗ must be lower semi-integrable (i.e., (f ∗)– dμ < ∞) since f ∗ is semi-integrable 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 semi-integrable, we assume the following for the rest of the proof. Assumption . f ∗ is upper semi-integrable. 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 semi-integrable, we have where (.) holds by (.), and (.) uses (.). Since f ∗ is upper semi-integrable and {Aik }k∈N is a σ -finite exhausting sequence, we have limk↑∞ Aik f ∗ dμ = f ∗ dμ < ∞. Thus applying limk↑∞ to the right-hand 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 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 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. 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Takashi Kamihigashi. 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, Journal of Inequalities and Applications, 2017, 24, DOI: 10.1186/s13660-016-1288-5