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Interchanging a limit and an integral: necessary and sufficient conditions
Journal of Inequalities and Applications volume 2020, Article number: 243 (2020)
Abstract
Let \(\{f_{n}\}_{n \in \mathbb {N}}\) be a sequence of integrable functions on a σfinite measure space \((\Omega, \mathscr {F}, \mu )\). Suppose that the pointwise limit \(\lim_{n \uparrow \infty } f_{n}\) exists μa.e. and is integrable. In this setting we provide necessary and sufficient conditions for the following equality to hold:
1 Introduction
Let \((\Omega, \mathscr {F}, \mu )\) be a measure space. Let \(\{f_{n}\}_{n \in \mathbb {N}}\) be a sequence of integrable functions \(f: \Omega \to \overline{\mathbb {R}} \equiv [\infty, \infty ]\) such that the pointwise limit \(\lim_{n \uparrow \infty } f_{n}\) exists μa.e. (μalmost everywhere) and is integrable. One often wishes to show that
In other words, one often wishes to interchange the limit \(\lim_{n \uparrow \infty }\) and the integral ∫. Various sufficient conditions for (1.1) are known, but only a small number of necessary and sufficient conditions are known in certain special cases. To our knowledge, no nontrivial necessary and sufficient condition for (1.1) applicable to the case that μ is σfinite is available in the literature.
One of the best known sufficient conditions for (1.1) is that there exists an integrable function g such that \(f_{n} \leq g\) for all \(n \in \mathbb {N}\). Under this condition, (1.1) holds by the dominated convergence theorem (e.g., [8, p. 89]). The condition is generalized in [9, p. 315] to the condition that there exist sequences \(\{\underline{f}_{n}\}\) and \(\{\overline{f}_{n}\}\) such that \(\underline{f}_{n} \leq f_{n} \leq \overline{f}_{n}\) for all \(n \in \mathbb {N}\) and such that both \(\{\underline{f}_{n}\}\) and \(\{\overline{f}_{n}\}\) satisfy (1.1); the same result is shown in [5] and [1, p. 134]. The condition is also necessary for (1.1) in a trivial way since it holds with \(\underline{f}_{n} = \overline{f}_{n} = f_{n}\) under (1.1).
It is shown in [7] that (1.1) holds if \(\Omega \subset \mathbb {R}\), μ is the Lebesgue measure, and the functions \(f_{n}\) and \(\lim_{n \uparrow \infty } f_{n}\) are probability densities on \(\mathbb {R}\). More generally, if μ is σfinite and the functions are nonnegative, a version of the Vitali convergence theorem provides a necessary and sufficient condition for (1.1) (see [8, p. 165]). In particular, uniform integrability of \(\{f_{n}\}\) (defined in Sect. 4) is equivalent to (1.1) provided that μ is σfinite and that \(f_{n} \geq 0\) for all \(n \in \mathbb {N}\). In the absence of this nonnegativity requirement, however, uniform integrability is only sufficient for (1.1), as shown in Sects. 4 and 5.
In the case that μ is finite, some necessary and sufficient conditions for (1.1) are given in [2, 3]. Those conditions are closely related, but not equivalent, to uniform integrability. Nevertheless, they are rarely mentioned in the current literature. We review one of the most general conditions in this case in Sect. 4.
In this paper we provide nontrivial necessary and sufficient conditions for (1.1) only under the assumption that μ is σfinite. We demonstrate by example that our conditions are strictly weaker than uniform integrability. We express them using the concept of a σfinite exhausting sequence, which is introduced in [4] to generalize Fatou’s lemma. A σfinite exhausting sequence is an increasing sequence of measurable sets of finite measure such that the complement of its entire union has measure 0. The basic idea of our conditions is to find a σfinite exhausting sequence such that (1.1) can be verified on each of the sets in the sequence and such that the sequence of integrals outside these sets can be controlled asymptotically.
The rest of the paper is organized as follows. In the next section we present our main results, which we prove in Sect. 3. In Sect. 4 we review some of the known results on (1.1) mentioned above. In Sect. 5 we present a simple example to which the known results do not apply, but our results easily apply.
2 Main results
Throughout the paper, we let \((\Omega, \mathscr {F}, \mu )\) be a measure space. Let \(\mathcal {L}\) be the set of measurable functions \(f: \Omega \rightarrow \overline{\mathbb {R}}\) (with \(\overline{\mathbb {R}}\) equipped with the Borel algebra). Let \(\mathcal {L}^{1}\) be the set of integrable functions in \(\mathcal {L}\). Let \(\mathcal {L}_{+}^{1} = \{f \in \mathcal {L}^{1}: f\geq 0\}\).
We say that a sequence \(\{A_{i}\}_{i \in \mathbb {N}}\) in \(\mathscr {F}\) is exhausting if
As in [4], we say that a sequence \(\{A_{i}\}_{i \in \mathbb {N}}\) in \(\mathscr {F}\) is a σfinite exhausting sequence if it is exhausting and
It is easy to see that μ is σfinite if and only if there exists a σfinite exhausting sequence. Thus whenever μ is σfinite, we have at least one σfinite exhausting sequence.
In what follows, by “\(f_{n} \to f\),” we mean “\(f_{n} \to f\) as \(n \uparrow \infty \).” We are ready to state our main results, which we prove in the next section.
Theorem 2.1
Suppose that μ is σfinite. Let \(\{f_{n}\}_{n \in \mathbb {N}}\) be a sequence in \(\mathcal {L}^{1}\) such that \(f_{n} \rightarrow f\) μa.e. for some \(f \in \mathcal {L}^{1}\). Then (1.1) holds if and only if there exists a σfinite exhausting sequence \(\{A_{i}\}_{i \in \mathbb {N}} \subset \mathscr {F}\) such that
A simple sufficient condition for (2.3) is uniform convergence on each \(A_{i}\); see (2.5) below. Condition (2.4) is somewhat similar to some of the conditions (such as uniform integrability) used in the known results reviewed in Sect. 4; see (4.3) and (4.5).
Theorem 2.2
Under the hypotheses of Theorem 2.1, (1.1) holds if and only if any σfinite exhausting sequence \(\{A_{i}\}_{i \in \mathbb {N}} \subset \mathscr {F}\) satisfying (2.3) also satisfies (2.4).
On the one hand, by Theorem 2.1, (1.1) can be established by constructing only one σfinite exhausting sequence satisfying both (2.3) and (2.4). On the other hand, by Theorem 2.2, (1.1) can be disproved by constructing only one σfinite exhausting sequence as well:
Corollary 2.1
Under the hypotheses of Theorem 2.1, (1.1) does not hold if there exists a σfinite exhausting sequence \(\{A_{i}\}_{i \in \mathbb {N}} \subset \mathscr {F}\) satisfying (2.3) but violating (2.4).
The following result shows that there always exists a σfinite exhausting sequence satisfying (2.3).
Lemma 2.1
Under the hypotheses of Theorem 2.1, there exists a σfinite exhausting sequence \(\{A_{i}\}_{i \in \mathbb {N}} \subset \mathscr {F}\) such that
Hence there exists a σfinite exhausting sequence \(\{A_{i}\}_{i \in \mathbb {N}} \subset \mathscr {F}\) satisfying (2.3).
Proof
Since μ is σfinite, there is a σfinite exhausting sequence \(\{B_{i}\}_{i \in \mathbb {N}} \subset \mathscr {F}\). Let \(\{\epsilon _{i}\}_{i \in \mathbb {N}}\) be a sequence in \((0,\infty )\) with \(\lim_{i \uparrow \infty } \epsilon _{i} = 0\). Let \(i \in \mathbb {N}\). Recall that \(f_{n} \rightarrow f\) μa.e.; thus 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 \(f_{n} \rightarrow f\) uniformly on \(E_{i}\).
For each \(i \in \mathbb {N}\), define
Then \(f_{n} \rightarrow f\) uniformly on \(A_{i}\). Since this is true for any \(i \in \mathbb {N}\), we obtain (2.5). By construction, we have
Hence it follows by [4, Lemma 8.4] that \(\{A_{i}\}\) is a σfinite exhausting sequence. Finally, the second conclusion holds since (2.5) implies (2.3). □
Lemma 2.1 is similar to Lusin’s version of Egorov’s theorem [6, p. 19], which shows that there exists a sequence \(\{A_{i}\}_{i \in \mathbb {N}} \subset \mathscr {F}\) such that \(\mu (\Omega \setminus \bigcup_{i \in \mathbb {N}} A_{i}) = 0\) and such that \(f_{n} \to f\) uniformly on \(A_{i}\) for each \(i \in \mathbb {N}\). Our result differs in that it explicitly shows that \(\mu (A_{i}) < \infty \) for each \(i \in \mathbb {N}\).
By Lemma 2.1, there exists a σfinite exhausting sequence satisfying (2.3). Given such a sequence, (1.1) is equivalent to (2.4):
Corollary 2.2
Under the hypotheses of Theorem 2.1, let \(\{A_{i}\}_{i \in \mathbb {N}} \subset \mathscr {F}\) be a σfinite exhausting sequence satisfying (2.3) (which exists by Lemma 2.1). Then (1.1) is equivalent to (2.4).
Proof
That (2.4) implies (1.1) is immediate from Theorem 2.1. The reverse implication follows from Theorem 2.2. □
3 Proof of Theorems 2.1 and 2.2
To simplify our arguments, we define
Whenever a sequence \(\{f_{n}\}\) from Φ is given, we let f be the μa.e. limit of \(\{f_{n}\}\) given in (3.1). With this convention, (1.1) can be written as
We also define
We prove Theorems 2.1 and 2.2 by showing that the following four conditions are equivalent.
Condition 3.1
\(\{f_{n}\} \in \Psi \).
Condition 3.2
There exists an exhausting sequence \(\{A_{i}\}_{i \in \mathbb {N}} \subset \mathscr {F}\) satisfying (2.4) such that
Condition 3.3
There exists a σfinite exhausting sequence \(\{A_{i}\}_{i \in \mathbb {N}} \subset \mathscr {F}\) satisfying (2.3) and (2.4).
Condition 3.4
Any σfinite exhausting sequence \(\{A_{i}\}_{i \in \mathbb {N}} \subset \mathscr {F}\) satisfying (2.3) also satisfies (2.4).
Note that Conditions 3.3 and 3.4 are the “if and only if” conditions in Theorems 2.1 and 2.2, respectively. The following result, which does not assume that μ is σfinite, shows that Conditions 3.1 and 3.2 are equivalent, but Condition 3.2 is necessary for Condition 3.1 only in a trivial way.
Lemma 3.1
Conditions 3.1and 3.2are equivalent.
Proof
Condition 3.1 implies Condition 3.2 with \(A_{i} = \Omega \) for all \(i \in \mathbb {N}\). Conversely, assume Condition 3.2. Let \(\{A_{i}\}_{i \in \mathbb {N}}\) be an exhausting sequence in \(\mathscr {F}\) satisfying (2.4) and (3.4). For any \(i, n \in \mathbb {N}\), we have
Applying \(\mathop {\overline {\lim }}_{n \uparrow \infty }\) and using (3.4), we have
Applying \(\mathop {\underline {\lim }}_{i \uparrow \infty }\) to the righthand side and recalling that f is integrable, we obtain
where the equality holds by integrability of f and (2.4).
Note from (3.5) that
Applying \(\mathop {\underline {\lim }}_{n \uparrow \infty }\) and then \(\mathop {\overline {\lim }}_{i \uparrow \infty }\), and using (2.4), we obtain
This together with (3.8) implies Condition 3.1. □
It is easy to see that Theorems 2.1 and 2.2 follow from the following result.
Theorem 3.1
Suppose that μ is σfinite. Then Conditions 3.1–3.4are equivalent.
Proof
We show this result by verifying the following chain of implications:
Lemma 3.1 shows that Condition 3.2 implies Condition 3.1. To see that Condition 3.3 implies Condition 3.2, fix \(i \in \mathbb {N}\) for the moment. Then (2.3) implies that there exists \(\eta > 0\) such that
Thus by the dominated convergence theorem, we obtain
Hence Condition 3.3 implies Condition 3.2. By Lemma 2.1, there exists a σfinite exhausting sequence \(\{A_{i}\}_{i \in \mathbb {N}} \subset \mathscr {F}\) satisfying (2.3). Thus Condition 3.4 implies Condition 3.3. We have verified the first three implications in (3.11). It remains to show the last implication.
To this end, assume Condition 3.1, which implies (3.2). Let \(\{A_{i}\}_{i \in \mathbb {N}}\) be any σfinite exhausting sequence satisfying (2.3). It suffices to verify (2.4). Applying \(\lim_{n \uparrow \infty }\) to (3.5) and recalling (3.13), we see that
where the limit on the righthand side exists since the limit on the lefthand side exists by (3.2). Since the lefthand side of (3.14) does not depend on i, we can apply \(\lim_{i \uparrow \infty }\) to the righthand side and use the integrability of f to get
Hence by (3.2) we have
Therefore
Thus (2.4) follows. This completes the proof. □
4 Known results
In this section we review some of the known results on (1.1) mentioned in Sect. 1. Throughout this section we take a sequence \(\{f_{n}\}_{n \in \mathbb {N}} \subset \mathcal {L}\) as given and assume the following.
Assumption 4.1
\(\{f_{n}\}_{n \in \mathbb {N}} \in \Phi \).
Essentially following [3], for \(i \in \mathbb {N}\) and \(\eta > 0\), define
In [3, p. 434] the following result is shown apparently under the implicit assumption that Ω is a subset of \(\mathbb {R}\) or \(\mathbb {R}^{n}\) of finite Lebesgue measure.
Theorem 4.1
Suppose that μ is finite. Then \(\{f_{n}\} \in \Psi \) if and only if
Proof
See [3, I]. □
The following result is a generalization of the dominated convergence theorem due to [9, p. 315].
Theorem 4.2
\(\{f_{n}\} \in \Psi \) if and only if there exist \(\{\underline{f}_{n}\}_{n\in \mathbb {N}}, \{\overline{f}_{n}\}_{n \in \mathbb {N}} \subset \Psi \) such that
Proof
The “only if” part trivially holds by setting \(\underline{f}_{n} = \overline{f}_{n} = f\) for all \(n \in \mathbb {N}\). For the “if” part, see [5] or [1, p. 134]. □
A sequence \(\{f_{n}\}_{n \in \mathbb {N}}\) in \(\mathcal {L}\) is called uniformly integrable ([8, p. 163]) if
See [8, Theorem 16.8] for various conditions equivalent to (4.5). The following result is an immediate implication of a version of the Vitali convergence theorem [8, Theorem 16.4].
Theorem 4.3
Suppose that μ is σfinite, and that \(\{f_{n}\} \subset \mathcal {L}_{+}^{1}\). Then \(\{f_{n}\} \in \Psi \) if and only if \(\{f_{n}\}\) is uniformly integrable.
Proof
This follows from [8, Lemma 16.4, Theorem 16.6] with \(p = 1\). □
Without the assumption that \(\{f_{n}\} \subset \mathcal {L}_{+}^{1}\), uniform integrability is still sufficient for (1.1), as shown in the next result. We prove it here for completeness; we do not claim originality.
Theorem 4.4
Suppose that μ is σfinite. If \(\{f_{n}\}\) is uniformly integrable, then \(\{f_{n}\} \in \Psi \).
Proof
Suppose that \(\{f_{n}\}\) is uniformly integrable. Then \(\{f_{n}\}\) is uniformly integrable, and \(\{f_{n}\} \in \Psi \) by Theorem 4.3. Since \(f_{n} \leq f_{n} \leq f_{n}\) for all \(n \in \mathbb {N}\), it follows by Theorem 4.2 that \(\{f_{n}\} \in \Psi \). □
In the next section we show by example that uniform integrability is not necessary for (1.1).
5 A simple example
In this section we present a simple example that satisfies (1.1) but violates (4.3) as well as uniform integrability. Let \(\Omega = \mathbb {R}_{+}\). Let \(\mathscr {F}\) be the σalgebra of Lebesgue measurable subsets of Ω. Let μ be the Lebesgue measure restricted to \(\mathscr {F}\). For \(n \in \mathbb {N}\), define \(f_{n} : \Omega \rightarrow \mathbb {R}\) by
Note that \(f_{n} \to 0\) pointwise and that \(\int f_{n} \, d\mu = 0\) for all \(n \in \mathbb {N}\). Thus (1.1) holds.
Since μ is not finite here, Theorem 4.1 simply does not apply. Furthermore, the “if and only if” condition in the theorem, (4.3), does not hold either, as we demonstrate here. For this purpose, let \(\eta > 0\), and let \(i \in \mathbb {N}\) be such that \(1/i < \eta < i\). Then
Since \(\int _{C(i,\eta )} f_{n} \, d\mu = n\) for any \(n \geq i\), we have
Hence the equality in (4.3) never holds for any \(\eta > 0\).
Let us now turn to the uniform integrability condition (4.5). Let \(h \in \mathcal {L}_{+}^{1}\) and \(n \in \mathbb {N}\). We have
where \mathbb{1}\{\cdot \} is the indicator function. Note that \(h(\omega ) \geq 1\) if \(h(\omega ) > n\) (≥1), and that \(h(\omega ) \geq 0\) otherwise; thus \mathbb{1}\{h(\omega )>n\}\le h(\omega ) for all \(\omega \in \Omega \). Hence
where the equality holds by integrability of h. From (5.4)–(5.6), we have
Since this is true for any \(h \in \mathcal {L}_{+}^{1}\), it follows that \(\{f_{n}\}\) is not uniformly integrable.
To see that our results apply even to this example, let \(A_{i} = [0, i)\) for \(i \in \mathbb {N}\). Then \(\{A_{i}\}_{i \in \mathbb {N}}\) is a σfinite exhausting sequence. Note that for each \(i \in \mathbb {N}\), we have \(f_{n} = f = 0\) on \(A_{i}\) for all \(n \geq i\). Thus both (2.3) and (3.4) trivially hold. For any \(i \in \mathbb {N}\), we have
Hence (2.4) also holds. It follows that both Conditions 3.2 and 3.3 hold, so that Theorems 2.1 and 3.1 apply here.
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Abbreviations
 a.e.:

almost everywhere
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Acknowledgements
The author would like to thank the anonymous reviewer for the constructive comments and suggestions, which helped significantly improve the presentation of the main results of the paper.
Funding
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. Interchanging a limit and an integral: necessary and sufficient conditions. J Inequal Appl 2020, 243 (2020). https://doi.org/10.1186/s1366002002502w
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DOI: https://doi.org/10.1186/s1366002002502w