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^{1}Email author
https://doi.org/10.1186/s13660-016-1288-5
© The Author(s) 2017
Received: 9 August 2016
Accepted: 22 December 2016
Published: 18 January 2017
Abstract
Keywords
1 Introduction
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 ‘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
\((\Omega, \mathscr {F}, \mu)\) 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.
2 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 semi-integrable if \(f^{+}\) or \(f^{-}\) is integrable, and upper (lower) semi-integrable if \(f^{+}\) (\(f^{-}\)) is integrable.
- (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)
3 A generalization of Fatou’s lemma
We are ready to state the main result of this paper.
Theorem 3.1
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 σ-finite exhausting sequence. Thus in Theorem 3.1, the requirement that \(\{A_{i}\}\) 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 \(\Omega= \mathbb {Z}_{+}\) and μ is the counting measure, we obtain a simple sufficient condition for the Fatou inequality:
Corollary 3.1
Proof
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 3.1.
Corollary 4.1
Let \(\{f_{n}\}_{n \in \mathbb {N}}\) be a sequence in \(\mathcal {L}(\Omega)\) such that for some upper semi-integrable function \(f \in \mathcal {L}(\Omega)\) we have \(f_{n} \leq f\) μ-a.e. for all \(n \in \mathbb {N}\). Then the Fatou inequality (1.1) holds.
Proof
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 semi-integrable. Then the Fatou inequality (1.1) holds.
Proof
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 \(\mathop {\overline {\lim }}_{n \uparrow\infty} f_{n}\) 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 equi-integrable. Suppose that \(\mathop {\overline {\lim }}_{n \uparrow\infty} f_{n}\) is semi-integrable. Then the Fatou inequality (1.1) holds.
Proof
5 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 σ-algebra 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
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
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 σ-finite 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 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 \(\int f_{n} \,d\mu= 0\) 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.
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 3.1. We start with some notation.
Assumption 6.1
We are ready to show our existence result.
Proposition 6.1
Proof
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
Proof
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
Example 7.2
Example 7.3
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.
8 Proof of Theorem 3.1
8.1 Preliminaries
Lemma 8.1
If \(f^{*}\) is not upper semi-integrable, then the Fatou inequality (1.1) holds.
Proof
Suppose that \(f^{*}\) is not upper semi-integrable. Then \(\int (f^{*})^{+} \,d\mu= \infty\), and \(f^{*}\) must be lower semi-integrable (i.e., \(\int(f^{*})^{-} \,d\mu< \infty\)) since \(f^{*}\) is semi-integrable 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 semi-integrable, 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
Proof
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 σ-finite exhausting sequence.
Proof
8.3 Completing the proof of Theorem 3.1
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 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.
9 Conclusions
Declarations
Acknowledgements
Financial support from the Japan Society for the Promotion of Science (JSPS KAKENHI Grant Number 15H05729) is gratefully acknowledged.
Open Access 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.
Authors’ Affiliations
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