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 {fn}n∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{f_{n}\}_{n \in \mathbb {N}}$\end{document} of measurable functions on a σ-finite measure space such that the integral of each fn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f_{n}$\end{document} as well as that of lim supn↑∞fn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\limsup_{n \uparrow\infty} f_{n}$\end{document} exists in R‾\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\overline{\mathbb {R}}$\end{document}, we provide a sufficient condition for the following inequality to hold: lim supn↑∞∫fndμ≤∫lim supn↑∞fndμ.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \limsup_{n \uparrow\infty} \int f_{n} \,d\mu\leq \int\limsup_{n \uparrow\infty} f_{n} \,d\mu. $$\end{document} 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 ( , F , μ) be a measure space. Let L( ) be the set of measurable functions f : → R. A standard version of (reverse) Fatou's lemma states that given a sequence {f n } n∈N in L( ), if there exists an integrable function f ∈ L( ) such that f n ≤ f μ-a.e. for all n ∈ N, then 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 f n as well as that of lim n↑∞ f n exists, 'uniform integrability' of {f + n } (where f + n is the positive part of f n ) is a sufficient condition for the Fatou inequality (.) in the case of a finite measure (e.g., [-]); so is 'equi-integrability' of the same sequence in the case of a σ -finite measure (see [, ]). These conditions are precisely defined in Section .
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.

Definitions
Given f ∈ L( ), let f + and fdenote the positive and negative parts of f , respectively; i.e., 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 {f n } n∈N of integrable functions in L( ) is called equi-integrable (e.g., [], page ) if the following conditions hold: (a) For any >  there exists δ >  such that any A ∈ F with μ(A) < δ satisfies

A generalization of Fatou's lemma
We are ready to state the main result of this paper.
Proof See Section .
It is shown in the proof (Lemma .) that (.) and (.) imply (.); i.e., (.) and (.) imply that {A i } i∈N is a σ -finite exhausting sequence. Thus in Theorem ., the requirement that {A i } 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: Corollary . Suppose that = Z + and that μ is the counting measure. Let {f n } n∈N be a sequence of semi-integrable functions in L( ) such that lim n↑∞ f n is semi-integrable. Suppose further that where the sum is understood as the Lebesgue integral with respect to the counting measure μ. Then Hence (.) follows from (.). Now (.) holds by Theorem ..

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 .. Proof Since f n ≤ f μ-a.e. for all n ∈ N and f is upper semi-integrable, f n is upper semiintegrable for each n ∈ N, and so is lim n↑∞ f n . For any σ -finite exhausting sequence where the equality holds by (.) since f is upper semi-integrable. Now the Fatou inequality (.) holds by Theorem .. Proof Recall that uniform integrability of {f + n } requires integrability of each f + n and condition (a) in Section  with f + n replacing f n . Let {A i } i∈N be any σ -finite exhausting sequence.
We have where the equality holds by condition (a) since {f + n } is uniformly integrable and lim i↑∞ μ( \ A i ) =  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 lim n↑∞ f n since they use the upper integral, which always exists, instead of the Lebesgue integral.
Fix i ∈ N for the moment. For each n ∈ N we have Applying sup n∈N to the leftmost and rightmost sides, we obtain The first supremum on the right-hand side converges to zero as The second supremum also converges to zero as i ↑ ∞ by (.)(ii) and condition (a) in Section . It follows that (.) holds for any sequence {A i } i∈N in F satisfying (.); thus by Theorem ., the Fatou inequality (.) holds.

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 {f + n }, does not apply either. Neither does Corollary . since equi-integrability implies uniform integrability on a finite measure space provided that sup n∈N |f n | dμ < ∞, which is the case here.
By contrast, Theorem . easily applies. To see this, note that, for each n ∈ N, f n is integrable, and so is lim n↑∞ f n . For i ∈ N, let Then {B i } i∈N is a σ -finite exhausting sequence. Let {A i } i∈N be any sequence in F satisfying (.)(i). For each fixed i ∈ N, for any n ≥ i, we have f n =  on B i , and \A i f n dμ = \B i f n dμ = . Thus the left-hand side of (.) is zero. Hence the Fatou inequality (.) holds by Theorem ..
In fact f n dμ =  for all n ∈ N, and lim n↑∞ f n = . Thus both sides of the Fatou inequality (.) equal zero.
In the next example, μ is not finite, and the sequence {f n } n∈N is uniformly bounded from below.

Example .
It is easy to see that there is no upper semi-integrable function that dominates {f n } n∈N ; thus Corollary . does not apply.

(.)
Thus {f + n } does not satisfy condition (a) in Section . To consider condition (b), let E ∈ F with μ(E) < ∞. Then which implies that lim n↑∞ μ(E ∩ [n, n + )) = . It follows that Hence {f + n } does not satisfy condition (b) either. Therefore {f + n } is far from being equiintegrable; as a consequence, Corollary . does not apply.
To see that Theorem . applies, note that, for each n ∈ N, f n is integrable for each n, and so is lim n↑∞ f n . For i ∈ N, let B i = [, i). Then {B i } i∈N is a σ -finite exhausting sequence. Take any sequence {A i } i∈N in F satisfying (.)(i). Then for each fixed i ∈ N we have \A i f n 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 f n dμ =  for all n ∈ N, and lim n↑∞ f n = ; thus both sides of the Fatou inequality (.) 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 .. We start with some notation.
For t ∈ Z + , let X t be a metric space. For t ∈ Z + , let t : X t → X t+ be a compact-valued upper hemicontinuous correspondence in the sense that, for each x ∈ X t , t (x) is a nonempty compact subset of X t+ , and for any convergent sequence {x n } n∈N in X t with limit x * ∈ X t and any sequence {y n } n∈N with y n ∈ t (x n ) for all n ∈ N, there exists a convergent subsequence {y n i } i∈N of {y n } n∈N with limit y * ∈ t (x * ); see [], page  and [], page , concerning this definition of upper hemicontinuity. For t ∈ Z + , let For t ∈ Z + , let r t : D t → R ∪ {-∞} be an upper semicontinuous function. Consider the following maximization problem: We say that a sequence {x t } ∞ t= is a feasible path (from x  ) if it satisfies (.). We say that a feasible path {x * t } ∞ t= is optimal (from x  ) if for any feasible path {x t } ∞ t= , we have where x *  = x  . For the above inequality to make sense, we assume the following.
In other words, the mapping r t (x t , We are ready to show our existence result. . By the definition of upper hemicontinuity, there exists a convergent subsequence of {x n j  } j∈N with limit x *  ∈  (x *  ). Continuing this way and using the diagonal argument, we see that there exists a subsequence of {{x n t } ∞ t= } n∈N , again denoted by To apply Corollary ., let f n (t) = r t (x n t , x n t+ ) for t ∈ Z + . By Assumption ., for each n ∈ N, f n (t) is an upper semi-integrable function of t ∈ Z + . For t ∈ Z + , let f * (t) = r t (x * t , x * t+ ). Since {x * t } ∞ 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 r t 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 We are ready to show that {x * t } ∞ t= is an optimal path. Recall from (.) that 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.

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: 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 + , X t = R + and 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 n t } ∞ t= for each n ∈ N bỹ Hence any f satisfying (.) must satisfy Since the right-hand side is not upper semi-integrable in t ∈ Z + by (. 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. Then the argument of Example . shows that an optimal path exists by Proposition ..

Proof of Theorem 3.1 8.1 Preliminaries
Throughout the proof, we fix {f n } n∈N and {B i } i∈N to be given by Theorem .. Define f * = lim n↑∞ f n . For n ∈ N, definef n = sup m≥n f m . We have The following observation helps to simplify the proof.
Lemma . If f * is not upper semi-integrable, then the Fatou inequality (.) 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.
where the last equality holds by (.) for {B i } and (.). It follows that {A i } satisfies (.).

Completing the proof of Theorem 3.1
Note from (.) that (f * ) + = lim n↑∞f + n . Let { i } i∈N be a sequence in R ++ such that lim i↑∞ i = . For each i ∈ N, by Egorov's theorem there exists E i ∈ F such that E i ⊂ B i , μ(B i \ E i ) < i , andf + n converges to (f * ) + uniformly on E i as n ↑ ∞. For i ∈ N, let Then, for each i ∈ N,f + n converges to (f * ) + uniformly on A i as n ↑ ∞. Thus (.) holds by Lemma ..
Note that {A i } i∈N satisfies (.) and (.) by construction. Thus by Lemma ., {A i } is a σ -finite exhausting sequence. Hence (.) holds by the hypothesis of Theorem .. Since (.) also holds as shown in the previous paragraph, the Fatou inequality (.) holds by Lemma ..

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 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 single-authored paper. The author read and approved the final manuscript.