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A generalization of Fatou’s lemma for extended realvalued functions on σfinite measure spaces: with an application to infinitehorizon optimization in discrete time
Journal of Inequalities and Applications volume 2017, Article number: 24 (2017)
Abstract
Given a sequence \(\{f_{n}\}_{n \in \mathbb {N}}\) of measurable functions on a σfinite measure space such that the integral of each \(f_{n}\) as well as that of \(\limsup_{n \uparrow\infty} f_{n}\) exists in \(\overline{\mathbb {R}}\), we provide a sufficient condition for the following inequality to hold:
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.
Introduction
Let \((\Omega, \mathscr {F}, \mu)\) be a measure space. Let \(\mathcal {L}(\Omega)\) be the set of measurable functions \(f: \Omega\rightarrow\overline{\mathbb {R}}\). A standard version of (reverse) Fatou’s lemma states that given a sequence \(\{f_{n}\}_{n \in \mathbb {N}}\) in \(\mathcal {L}(\Omega)\), if there exists an integrable function \(f \in \mathcal {L}(\Omega)\) such that \(f_{n} \leq f\) μa.e. for all \(n \in \mathbb {N}\), then
where \(\mathop {\overline {\lim }}= \limsup\). 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 \(\mathop {\overline {\lim }}_{n \uparrow\infty} f_{n}\) exists, ‘uniform integrability’ of \(\{f_{n}^{+}\}\) (where \(f_{n}^{+}\) is the positive part of \(f_{n}\)) is a sufficient condition for the Fatou inequality (1.1) in the case of a finite measure (e.g., [1–4]); so is ‘equiintegrability’ of the same sequence in the case of a σfinite 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
\((\Omega, \mathscr {F}, \mu)\) is a σfinite 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 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., [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 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 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 infinitehorizon deterministic optimization problems in discrete time. In Section 8 we prove our main result.
Definitions
Given \(f \in \mathcal {L}(\Omega)\), let \(f^{+}\) and \(f^{}\) denote the positive and negative parts of f, respectively; i.e., \(f^{+} = \max\{f, 0\}\) and \(f^{} = \max\{f, 0\}\). A function \(f \in \mathcal {L}(\Omega)\) is called semiintegrable if \(f^{+}\) or \(f^{}\) is integrable, and upper (lower) semiintegrable if \(f^{+}\) (\(f^{}\)) is integrable.
We say that a sequence \(\{A_{i}\}_{i \in \mathbb {N}}\) in \(\mathscr {F}\) is a σfinite exhausting sequence if
It is easy to see that μ is σfinite if and only if there exists a σfinite exhausting sequence. Since we assume that μ is σfinite, we have at least one σfinite exhausting sequence.
A sequence \(\{f_{n}\}_{n \in \mathbb {N}}\) of integrable functions in \(\mathcal {L}(\Omega)\) is called equiintegrable (e.g., [6], page 16) if the following conditions hold:

(a)
For any \(\epsilon> 0\) there exists \(\delta> 0\) such that any \(A \in \mathscr {F}\) with \(\mu(A) < \delta\) satisfies
$$\begin{aligned} \sup_{n \in \mathbb {N}} \int_{A} \vert f_{n}\vert \,d\mu\leq\epsilon. \end{aligned}$$(2.3) 
(b)
For any \(\epsilon> 0\) there exists \(E \in \mathscr {F}\) with \(\mu(E) < \infty\) such that
$$\begin{aligned} \sup_{n \in \mathbb {N}} \int_{\Omega\setminus E} \vert f_{n}\vert \,d\mu\leq\epsilon. \end{aligned}$$(2.4)
Suppose that \(\mu(\Omega) < \infty\). A sequence \(\{f_{n}\}_{n\in \mathbb {N}}\) of integrable functions in \(\mathcal {L}(\Omega)\) is called uniformly integrable (e.g., [3], page 144) if
It is well known that a sequence \(\{f_{n}\}_{n \in \mathbb {N}}\) of integrable functions in \(\mathcal {L}(\Omega)\) is uniformly integrable if and only if \(\sup_{n \in \mathbb {N}} \int \vert f_{n}\vert \,d\mu< \infty\) 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 equiintegrability. Conversely, provided that \(\sup_{n \in \mathbb {N}} \int \vert f_{n}\vert \,d\mu< \infty\), equiintegrability 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 \(\{f_{n}\}_{n \in \mathbb {N}}\) be a sequence of semiintegrable functions in \(\mathcal {L}(\Omega)\) such that \(\mathop {\overline {\lim }}_{n \uparrow\infty} f_{n}\) is semiintegrable. Let \(\{B_{i}\}_{i \in \mathbb {N}} \subset \mathscr {F}\) be a σfinite exhausting sequence. Suppose that
for any σfinite exhausting sequence \(\{A_{i}\}_{i \in \mathbb {N}} \subset \mathscr {F}\) 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 \(\{A_{i}\} _{i \in \mathbb {N}}\) is a σfinite exhausting sequence. Thus in Theorem 3.1, the requirement that \(\{A_{i}\}\) be a σfinite 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 \(\Omega= \mathbb {Z}_{+}\) and μ is the counting measure, we obtain a simple sufficient condition for the Fatou inequality:
Corollary 3.1
Suppose that \(\Omega= \mathbb {Z}_{+}\) and that μ is the counting measure. Let \(\{f_{n}\}_{n \in \mathbb {N}}\) be a sequence of semiintegrable functions in \(\mathcal {L}(\Omega)\) such that \(\mathop {\overline {\lim }}_{n \uparrow\infty} f_{n}\) is semiintegrable. Suppose further that
where the sum is understood as the Lebesgue integral with respect to the counting measure μ. Then
Proof
Assume (3.3). For \(i \in \mathbb {N}\), let \(B_{i} = \{0, \ldots, i1\}\). Then \(\{B_{i}\}_{i \in \mathbb {N}}\) is a σfinite exhausting sequence. Let \(\{A_{i}\}_{i \in \mathbb {N}} \subset \mathscr {F}\) satisfy (3.2). Then \(A_{i} = B_{i}\) 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 \(\{f_{n}\}_{n \in \mathbb {N}}\) be a sequence in \(\mathcal {L}(\Omega)\) such that for some upper semiintegrable function \(f \in \mathcal {L}(\Omega)\) we have \(f_{n} \leq f\) μa.e. for all \(n \in \mathbb {N}\). Then the Fatou inequality (1.1) holds.
Proof
Since \(f_{n} \leq f\) μa.e. for all \(n \in \mathbb {N}\) and f is upper semiintegrable, \(f_{n}\) is upper semiintegrable for each \(n \in \mathbb {N}\), and so is \(\mathop {\overline {\lim }}_{n \uparrow\infty} f_{n}\). For any σfinite exhausting sequence \(\{A_{i}\}_{i \in \mathbb {N}}\) we have
where the equality holds by (2.2) since f is upper semiintegrable. 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 \(\mu(\Omega) < \infty\). Let \(\{f_{n}\}_{n \in \mathbb {N}}\) be a sequence of functions in \(\mathcal {L}(\Omega)\) such that \(\{f_{n}^{+}\}_{n \in \mathbb {N}}\) is uniformly integrable. Suppose further that \(\mathop {\overline {\lim }}_{n \uparrow\infty} f_{n}\) is semiintegrable. Then the Fatou inequality (1.1) holds.
Proof
Recall that uniform integrability of \(\{f_{n}^{+}\}\) requires integrability of each \(f_{n}^{+}\) and condition (a) in Section 2 with \(f_{n}^{+}\) replacing \(f_{n}\). Let \(\{A_{i}\}_{i \in \mathbb {N}}\) be any σfinite exhausting sequence. We have
where the equality holds by condition (a) since \(\{f_{n}^{+}\}\) is uniformly integrable and \(\lim_{i \uparrow\infty} \mu(\Omega \setminus A_{i}) = 0\) 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 [5], Lemma 3.3 and [6], Corollary 3.3. The latter results (unlike Corollary 4.3 below) do not require upper semiintegrability of \(\mathop {\overline {\lim }}_{n \uparrow\infty} f_{n}\) since they use the upper integral, which always exists, instead of the Lebesgue integral.
Corollary 4.3
Let \(\{f_{n}\}_{n \in \mathbb {N}}\) be a sequence of integrable functions in \(\mathcal {L}(\Omega)\) such that \(\{f_{n}^{+}\}_{n \in \mathbb {N}}\) is equiintegrable. Suppose that \(\mathop {\overline {\lim }}_{n \uparrow\infty} f_{n}\) is semiintegrable. Then the Fatou inequality (1.1) holds.
Proof
By equiintegrability of \(\{f_{n}^{+}\}\) and condition (b) in Section 2, there exists a sequence \(\{E_{i}\}_{i \in \mathbb {N}}\) in \(\mathscr {F}\) such that \(\mu(E_{i}) < \infty\) for all \(i \in \mathbb {N}\) and
Since μ is σfinite, there exists a σfinite exhausting sequence \(\{C_{i}\}_{i \in \mathbb {N}}\). For \(i \in \mathbb {N}\), let \(B_{i} = (\bigcup_{j=1}^{i} E_{j}) \cup C_{i}\). Then \(\{B_{i}\}_{i \in \mathbb {N}}\) is also a σfinite exhausting sequence. Let \(\{A_{i}\}_{i \in \mathbb {N}}\) be a sequence in \(\mathscr {F}\) satisfying (3.2).
Fix \(i \in \mathbb {N}\) for the moment. For each \(n \in \mathbb {N}\) we have
Applying \(\sup_{n \in \mathbb {N}}\) to the leftmost and rightmost sides, we obtain
The first supremum on the righthand side converges to zero as \(i \uparrow\infty\) by (4.3) since \(E_{i} \subset B_{i}\) for all \(i \in \mathbb {N}\). The second supremum also converges to zero as \(i \uparrow\infty\) by (3.2)(ii) and condition (a) in Section 2. It follows that (3.1) holds for any sequence \(\{A_{i}\}_{i \in \mathbb {N}}\) in \(\mathscr {F}\) 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 \(\mathbb {R}\). Accordingly, \(\mathscr {F}\) is taken to be the σalgebra of Lebesgue measurable subsets of Ω, and μ the Lebesgue measure restricted to \(\mathscr {F}\).
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 \(\Omega= [1, 1] \setminus\{0\}\). For \(n \in \mathbb {N}\), define \(f_{n} : \Omega\rightarrow \mathbb {R}\) by
It is easy to see that there is no upper semiintegrable function that dominates \(\{f_{n}\}_{n \in \mathbb {N}}\); thus Corollary 4.1 does not apply. Furthermore, \(\{f_{n}^{+}\}\) is not uniformly integrable; indeed, for any \(c \geq0\) we have
Hence Corollary 4.2, which requires uniform integrability of \(\{f_{n}^{+}\}\), does not apply either. Neither does Corollary 4.3 since equiintegrability implies uniform integrability on a finite measure space provided that \(\sup_{n \in \mathbb {N}} \int \vert f_{n}\vert \,d\mu< \infty\), which is the case here.
By contrast, Theorem 3.1 easily applies. To see this, note that, for each \(n \in \mathbb {N}\), \(f_{n}\) is integrable, and so is \(\lim_{n \uparrow\infty} f_{n}\). For \(i \in \mathbb {N}\), let
Then \(\{B_{i}\}_{i \in \mathbb {N}}\) is a σfinite exhausting sequence. Let \(\{A_{i}\}_{i \in \mathbb {N}}\) be any sequence in \(\mathscr {F}\) satisfying (3.2)(i). For each fixed \(i \in \mathbb {N}\), for any \(n \geq i\), we have \(f_{n} = 0\) on \(B_{i}\), and \(\int_{\Omega\setminus A_{i}} f_{n} \,d\mu= \int_{\Omega\setminus B_{i}} f_{n} \,d\mu= 0\). Thus the lefthand side of (3.1) is zero. Hence the Fatou inequality (1.1) holds by Theorem 3.1.
In fact \(\int f_{n} \,d\mu= 0\) for all \(n \in \mathbb {N}\), and \(\lim_{n \uparrow\infty} f_{n} = 0\). Thus both sides of the Fatou inequality (1.1) equal zero.
In the next example, μ is not finite, and the sequence \(\{f_{n}\} _{n \in \mathbb {N}}\) is uniformly bounded from below.
Example 5.2
Let \(\Omega= \mathbb {R}_{+}\). For \(n \in \mathbb {N}\), define \(f_{n} : \Omega \rightarrow \mathbb {R}\) by
It is easy to see that there is no upper semiintegrable function that dominates \(\{f_{n}\}_{n \in \mathbb {N}}\); thus Corollary 4.1 does not apply.
For any \(\delta\in(0,1)\) we have
Thus \(\{f_{n}^{+}\}\) does not satisfy condition (a) in Section 2. To consider condition (b), let \(E \in \mathscr {F}\) with \(\mu(E) < \infty\). Then
which implies that \(\lim_{n \uparrow\infty} \mu(E \cap[n,n+1)) = 0\). It follows that
Hence \(\{f_{n}^{+}\}\) does not satisfy condition (b) either. Therefore \(\{f_{n}^{+}\}\) is far from being equiintegrable; as a consequence, Corollary 4.3 does not apply.
To see that Theorem 3.1 applies, note that, for each \(n \in \mathbb {N}\), \(f_{n}\) is integrable for each n, and so is \(\lim_{n \uparrow \infty} f_{n}\). For \(i \in \mathbb {N}\), let \(B_{i} = [0, i)\). Then \(\{B_{i}\} _{i \in \mathbb {N}}\) is a σfinite exhausting sequence. Take any sequence \(\{A_{i}\}_{i \in \mathbb {N}}\) in \(\mathscr {F}\) satisfying (3.2)(i). Then for each fixed \(i \in \mathbb {N}\) we have \(\int_{\Omega \setminus A_{i}} f_{n} \,d\mu= 0\) for all \(n \geq i\). Thus the lefthand 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 \(\int f_{n} \,d\mu= 0\) for all \(n \in \mathbb {N}\), and \(\lim_{n \uparrow\infty} f_{n} = 0\); thus both sides of the Fatou inequality (1.1) equal zero.
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 3.1. We start with some notation.
For \(t \in \mathbb {Z}_{+}\), let \(X_{t}\) be a metric space. For \(t \in \mathbb {Z}_{+}\), let \(\Gamma_{t} : X_{t} \rightarrow X_{t+1}\) be a compactvalued upper hemicontinuous correspondence in the sense that, for each \(x \in X_{t}\), \(\Gamma_{t}(x)\) is a nonempty compact subset of \(X_{t+1}\), and for any convergent sequence \(\{x_{n}\}_{n \in \mathbb {N}}\) in \(X_{t}\) with limit \(x^{*} \in X_{t}\) and any sequence \(\{y_{n}\}_{n \in \mathbb {N}}\) with \(y_{n} \in\Gamma_{t}(x_{n})\) for all \(n \in \mathbb {N}\), there exists a convergent subsequence \(\{y_{n_{i}}\}_{i \in \mathbb {N}}\) of \(\{ y_{n}\}_{n \in \mathbb {N}}\) with limit \(y^{*} \in\Gamma_{t}(x^{*})\); see [16], page 56 and [17], page 564, concerning this definition of upper hemicontinuity. For \(t \in \mathbb {Z}_{+}\), let
For \(t \in \mathbb {Z}_{+}\), let \(r_{t} : D_{t} \rightarrow \mathbb {R}\cup\{\infty\} \) be an upper semicontinuous function.
Consider the following maximization problem:
We say that a sequence \(\{x_{t}\}_{t=1}^{\infty}\) is a feasible path (from \(x_{0}\)) if it satisfies (6.3). We say that a feasible path \(\{x^{*}_{t}\} _{t=1}^{\infty}\) is optimal (from \(x_{0}\)) if for any feasible path \(\{x_{t}\}_{t=1}^{\infty}\), we have
where \(x^{*}_{0} = x_{0}\). For the above inequality to make sense, we assume the following.
Assumption 6.1
For each feasible path \(\{x_{t}\}_{t=1}^{\infty}\), we have
In other words, the mapping \(r_{t}(x_{t}, x_{t+1}): t \mapsto \mathbb {R}\cup\{ \infty\}\) is upper semiintegrable.
We are ready to show our existence result.
Proposition 6.1
Let Assumption 6.1 hold. Suppose that, for any sequence \(\{ \{x_{t}^{n}\}_{t=1}^{\infty}\}_{n \in \mathbb {N}}\) of feasible paths, we have
Then there exists an optimal path.
Proof
Let
where the supremum is taken over all feasible paths \(\{x_{t}\} _{t=1}^{\infty}\). By the definition of ν, there exists a sequence \(\{\{x_{t}^{n}\}_{t=1}^{\infty}\}_{n \in \mathbb {N}}\) of feasible paths such that
Since \(\Gamma_{0}(x_{0})\) is compact, there exists a convergent subsequence \(\{x_{1}^{n_{j}}\}_{j \in \mathbb {N}}\) of \(\{x_{1}^{n}\}_{n \in \mathbb {N}}\) with limit \(x^{*}_{1} \in\Gamma_{0}(x_{0})\). By the definition of upper hemicontinuity, there exists a convergent subsequence of \(\{ x_{2}^{n_{j}}\}_{j \in \mathbb {N}}\) with limit \(x^{*}_{2} \in\Gamma _{1}(x^{*}_{1})\). Continuing this way and using the diagonal argument, we see that there exists a subsequence of \(\{\{x_{t}^{n}\} _{t=1}^{\infty}\}_{n \in \mathbb {N}}\), again denoted by \(\{\{x_{t}^{n}\} _{t=1}^{\infty}\}_{n \in \mathbb {N}}\), such that, for each \(t \in \mathbb {N}\), \(x_{t}^{n}\) converges to some \(x^{*}_{t}\) as \(n \uparrow\infty\), and for each \(t \in \mathbb {Z}_{+}\), \(x^{*}_{t+1} \in\Gamma_{t}(x^{*}_{t})\). Hence \(\{x^{*}_{t}\}_{t=1}^{\infty}\) is a feasible path, which implies that
To apply Corollary 3.1, let \(f_{n}(t) = r_{t}(x_{t}^{n}, x_{t+1}^{n})\) for \(t \in \mathbb {Z}_{+}\). By Assumption 6.1, for each \(n \in \mathbb {N}\), \(f_{n}(t)\) is an upper semiintegrable function of \(t \in \mathbb {Z}_{+}\). For \(t \in \mathbb {Z}_{+}\), let \(f^{*}(t) = r_{t}(x^{*}_{t}, x^{*}_{t+1})\). Since \(\{x^{*}_{t}\}_{t=1}^{\infty}\) is feasible as shown above, \(f^{*}(t)\) is also an upper semiintegrable function of \(t \in \mathbb {Z}_{+}\) by Assumption 6.1. For each \(t \in \mathbb {Z}_{+}\), by upper semicontinuity of \(r_{t}\) we have
Since the rightmost side is an upper semiintegrable function of \(t \in \mathbb {Z}_{+}\), 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 \(\{x^{*}_{t}\}_{t=1}^{\infty}\) 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 \(\{x^{*}_{t}\}_{t=1}^{\infty}\) 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 \(\overline{f} : \mathbb {Z}_{+} \rightarrow \mathbb {R}_{+}\) such that, for any feasible path \(\{x_{t}\} _{t=1}^{\infty}\), we have
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 \(\{\{x_{t}^{n}\}_{t=1}^{\infty}\}_{n \in \mathbb {N}}\) of feasible paths. Let \(\{\{x_{t}^{n}\}_{t=1}^{\infty}\}_{n \in \mathbb {N}}\) 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 \(u : \mathbb {R}_{+} \rightarrow \mathbb {R}\cup\{\infty\}\) be a strictly increasing, upper semicontinuous function. Let \(\delta: \mathbb {R}_{+} \rightarrow \mathbb {R}_{++}\) 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 (6.2)(6.4) such that, for all \(t \in \mathbb {Z}_{+}\), \(X_{t} = \mathbb {R}_{+}\) and
It is easy to see from (7.2) that
For simplicity, we assume that there exists \(\theta> 0\) 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 discretetime economic models assume an exponential discount function of the form
for some \(\beta\in(0,1)\). In this case, Corollary 6.1 easily applies. To see this, let \(\overline{f}(t) = \beta^{t} u(x_{0})\) for \(t \in \mathbb {Z}_{+}\). Then \(\overline{f} : \mathbb {Z}_{+} \rightarrow \mathbb {R}_{+}\) 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 \(\alpha> 0\).
In this example, Corollary 6.1 does not apply since there exists no integrable function \(\overline{f} : \mathbb {Z}_{+} \rightarrow \mathbb {R}_{+}\) satisfying (6.16) for all feasible paths. To see this, define the feasible path \(\{\tilde{x}_{t}^{n}\}_{t=1}^{\infty}\) for each \(n \in \mathbb {N}\) by
Then
Hence any f̅ satisfying (6.16) must satisfy
Since the righthand side is not upper semiintegrable in \(t \in \mathbb {Z}_{+}\) 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 \(\{ \{x_{t}^{n}\}_{t=1}^{\infty}\}_{n \in \mathbb {N}}\) be a sequence of feasible paths. For any \(n, i \in \mathbb {N}\) 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 \(\{f_{n}\}_{n \in \mathbb {N}}\) and \(\{B_{i}\}_{i \in \mathbb {N}}\) to be given by Theorem 3.1. Define \(f^{*} = \mathop {\overline {\lim }}_{n \uparrow\infty} f_{n}\). For \(n \in \mathbb {N}\), define \(\hat {f}_{n} = \sup_{m \geq n} f_{m}\). We have
The following observation helps to simplify the proof.
Lemma 8.1
If \(f^{*}\) is not upper semiintegrable, then the Fatou inequality (1.1) holds.
Proof
Suppose that \(f^{*}\) is not upper semiintegrable. Then \(\int (f^{*})^{+} \,d\mu= \infty\), and \(f^{*}\) must be lower semiintegrable (i.e., \(\int(f^{*})^{} \,d\mu< \infty\)) since \(f^{*}\) is semiintegrable by hypothesis. It follows that \(\int f^{*} \,d\mu= \int(f^{*})^{+} \,d\mu \int(f^{*})^{} \,d\mu= \infty\). 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 semiintegrable.
Lemmas
We establish three lemmas before completing the proof of Theorem 3.1.
Lemma 8.2
Suppose that there exists a σfinite exhausting sequence \(\{ A_{i}\}_{i \in \mathbb {N}}\) satisfying (3.1) and the following:
Then the Fatou inequality (1.1) holds.
Proof
Since each \(f_{n}\) is semiintegrable, we have
By (3.1) there exists a subsequence \(\{A_{i_{k}}\}_{k \in \mathbb {N}}\) of \(\{A_{i}\}_{i \in \mathbb {N}}\) such that
Note that \(\{A_{i_{k}}\}_{k \in \mathbb {N}}\) is a σfinite exhausting sequence.
Fix \(k \in \mathbb {N}\) for the moment. Replacing i with \(i_{k}\) in (8.3) and applying \(\mathop {\overline {\lim }}_{n \uparrow\infty}\) to both sides of the resulting equation, we obtain
where (8.7) holds by (8.4), and (8.8) uses (8.2).
Since \(f^{*}\) is upper semiintegrable and \(\{A_{i_{k}}\}_{k \in \mathbb {N}}\) is a σfinite exhausting sequence, we have \(\lim_{k \uparrow \infty} \int_{A_{i_{k}}} f^{*} \,d\mu= \int f^{*} \,d\mu< \infty\). Thus applying \(\lim_{k \uparrow\infty}\) to the righthand side of (8.8) yields
where the last inequality uses (8.5). The Fatou inequality (1.1) follows. □
Lemma 8.3
Let \(\{A_{i}\}_{i \in \mathbb {N}}\) be a sequence in \(\mathscr {F}\) such that, for each \(i \in \mathbb {N}\), \(\mu(A_{i}) < \infty\) and \(\hat{f}_{n}^{+}\) converges to \((f^{*})^{+}\) uniformly on \(A_{i}\) as \(n \uparrow\infty\). Then \(\{ A_{i}\}_{i \in \mathbb {N}}\) satisfies (8.2).
Proof
Let \(i \in \mathbb {N}\). Let \(\delta> 0\). Since \(\hat{f}_{n}^{+}\) converges to \((f^{*})^{+}\) uniformly on \(A_{i}\) as \(n \uparrow\infty\), for sufficiently large \(n \in \mathbb {N}\) we have \(f_{n} \leq\hat{f}_{n} \leq \hat{f}_{n}^{+} \leq(f^{*})^{+} + \delta\) on \(A_{i}\). Since \((f^{*})^{+}\) is integrable by Assumption 8.1 and \(\mu (A_{i}) < \infty\), (8.2) holds by Fatou’s lemma. □
Lemma 8.4
Let \(\{A_{i}\}_{i \in \mathbb {N}}\) be a sequence in \(\mathscr {F}\) satisfying (2.1) and (3.2). Then \(\{A_{i}\}\) is a σfinite exhausting sequence.
Proof
Since \(\{A_{i}\}\) satisfies (2.1) by hypothesis, it suffices to verify (2.2). For any \(i, j \in \mathbb {N}\) with \(i \leq j\), by (2.1) for \(\{B_{i}\}\), we have
where the convergence holds by (3.2). It follows that
Therefore
Since \(\bigcup_{i \in \mathbb {N}} A_{i} \subset\bigcup_{i \in \mathbb {N}} B_{i}\), we have
where the last equality holds by (2.2) for \(\{B_{i}\}\) and (8.12). It follows that \(\{A_{i}\}\) satisfies (2.2). □
Completing the proof of Theorem 3.1
Note from (8.1) that \((f^{*})^{+} = \lim_{n \uparrow \infty} \hat{f}_{n}^{+}\). Let \(\{\epsilon_{i}\}_{i \in \mathbb {N}}\) be a sequence in \(\mathbb {R}_{++}\) such that \(\lim_{i \uparrow\infty} \epsilon _{i} = 0\). For each \(i \in \mathbb {N}\), by Egorov’s theorem there exists \(E_{i} \in \mathscr {F}\) such that \(E_{i} \subset B_{i}\), \(\mu(B_{i} \setminus E_{i}) < \epsilon_{i}\), and \(\hat{f}_{n}^{+}\) converges to \((f^{*})^{+}\) uniformly on \(E_{i}\) as \(n \uparrow\infty\). For \(i \in \mathbb {N}\), let
Then, for each \(i \in \mathbb {N}\), \(\hat{f}_{n}^{+}\) converges to \((f^{*})^{+}\) uniformly on \(A_{i}\) as \(n \uparrow\infty\). Thus (8.2) holds by Lemma 8.3.
Note that \(\{A_{i}\}_{i \in \mathbb {N}}\) satisfies (2.1) and (3.2) by construction. Thus by Lemma 8.4, \(\{A_{i}\} \) is a σfinite 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 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.
References
Chow, YS, Robbins, H, Siegmund, D: Great Expectations: The Theory of Optimal Stopping. Houghton Mifflin, Boston (1971)
Cairoli, R, Dalang, RC: Sequential Stochastic Optimization. Wiley, New York (1996)
Denkowski, Z, Migórski, S, Papageorgiou, NS: An Introduction to Nonlinear Analysis: Theory. Springer, New York (2003)
Shiryaev, AN: Optimal Stopping Rules. Springer, Berlin (2008)
Giner, E: Calmness properties and contingent subgradients of integral functions on Lebesgue spaces \(L_{p}\), \(1 \leq p < \infty\). SetValued Anal. 17, 223243 (2009)
Giner, E: Fatou’s lemma and lower epilimits of integral functions. J. Math. Anal. Appl. 394, 1329 (2012)
Kamihigashi, T: Necessity of transversality conditions for infinite horizon problems. Econometrica 69, 9951012 (2001)
Kamihigashi, T: Necessity of transversality conditions for stochastic problems. J. Econ. Theory 109, 140149 (2003)
Kamihigashi, T: Necessity of transversality conditions for stochastic models with bounded or CRRA utility. J. Econ. Dyn. Control 29, 13131329 (2005)
Kamihigashi, T: Elementary results on solutions to the Bellman equation of dynamic programming: existence, uniqueness, and convergence. Econ. Theory 56, 251273 (2014)
Cesari, L: OptimizationTheory and Applications. Springer, New York (1983)
Ekeland, I, Scheinkman, JA: Transversality conditions for some infinite horizon discrete time optimization problems. Math. Oper. Res. 11, 216229 (1986)
Brezis, H, Lieb, E: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88, 486490 (1983)
Balder, EJ: A unifying note on Fatou’s lemma in several dimensions. Math. Oper. Res. 9, 267275 (1984)
Loeb, PA, Sun, Y: A general Fatou lemma. Adv. Math. 213, 741762 (2007)
Stokey, N, Lucas, RE Jr.: Recursive Methods in Economic Dynamics. Harvard University Press, Cambridge (1989)
Aliprantis, CD, Border, KC: Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd edn. Springer, Berlin (2006)
Le Van, C, Morhaim, L: Optimal growth models with bounded or unbounded returns: a unifying approach. J. Econ. Theory 105, 158187 (2002)
Zarr, N, Alexander, WH, Brown, JW: Discounting of reward sequences: a test of competing formal models of hyperbolic discounting. Front. Psychol. 5, 19 (2014)
Acknowledgements
Financial support from the Japan Society for the Promotion of Science (JSPS KAKENHI Grant Number 15H05729) is gratefully acknowledged.
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Kamihigashi, T. A generalization of Fatou’s lemma for extended realvalued functions on σfinite measure spaces: with an application to infinitehorizon optimization in discrete time. J Inequal Appl 2017, 24 (2017). https://doi.org/10.1186/s1366001612885
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DOI: https://doi.org/10.1186/s1366001612885
Keywords
 Fatou’s lemma
 σfinite measure space
 infinitehorizon optimization
 hyperbolic discounting
 existence of optimal paths