The following result is due to Beppo Levi, who proved a slight generalization in 1906 of an earlier result by Henri Lebesgue. In what follows, denotes the -algebra of Borel sets on . By definition, contains the set and all Borel subsets of
Theorem
Let be a measure space, and . Consider a pointwise non-decreasing sequence of -measurable non-negative functions , i.e., for every and every ,
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Set the pointwise limit of the sequence to be . That is, for every ,
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Then is -measurable and
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Remark 1. The integrals may be finite or infinite.
Remark 2. The theorem remains true if its assumptions hold -almost everywhere. In other words, it is enough that there is a null set such that the sequence non-decreases for every To see why this is true, we start with an observation that allowing the sequence to pointwise non-decrease almost everywhere causes its pointwise limit to be undefined on some null set . On that null set, may then be defined arbitrarily, e.g. as zero, or in any other way that preserves measurability. To see why this will not affect the outcome of the theorem, note that since we have, for every
- and
provided that is -measurable.: section 21.38 (These equalities follow directly from the definition of the Lebesgue integral for a non-negative function).
Remark 3. Under the assumptions of the theorem,
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(Note that the second chain of equalities follows from Remark 5).
Remark 4. The proof below does not use any properties of the Lebesgue integral except those established here. The theorem, thus, can be used to prove other basic properties, such as linearity, pertaining to Lebesgue integration.
Remark 5 (monotonicity of the Lebesgue integral). In the proof below, we apply the monotonic property of the Lebesgue integral to non-negative functions only. Specifically (see Remark 4), let the functions be -measurable.
- If everywhere on then
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- If and then
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Proof. Denote by the set of simple -measurable functions such that everywhere on
1. Since we have
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By definition of the Lebesgue integral and the properties of supremum,
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2. Let be the indicator function of the set It can be deduced from the definition of the Lebesgue integral that
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if we notice that, for every outside of Combined with the previous property, the inequality implies
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Proof
This proof does not rely on Fatou's lemma; however, we do explain how that lemma might be used. Those not interested in this independency of the proof may skip the intermediate results below.
Lebesgue integral as measure
Lemma 1. Let be a measurable space. Consider a simple -measurable non-negative function . For a subset , define
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Then is a measure on .
Proof
Monotonicity follows from Remark 5. Here, we will only prove countable additivity, leaving the rest up to the reader. Let , where all the sets are pairwise disjoint. Due to simplicity,
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for some finite non-negative constants and pairwise disjoint sets such that . By definition of the Lebesgue integral,
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Since all the sets are pairwise disjoint, the countable additivity of gives us
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Since all the summands are non-negative, the sum of the series, whether this sum is finite or infinite, cannot change if summation order does. For that reason,
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as required.
"Continuity from below"
The following property is a direct consequence of the definition of measure.
Lemma 2. Let be a measure, and , where
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is a non-decreasing chain with all its sets -measurable. Then
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Proof of theorem
Step 1. We begin by showing that is –measurable.: section 21.3
Note. If we were using Fatou's lemma, the measurability would follow easily from Remark 3(a).
To do this without using Fatou's lemma, it is sufficient to show that the inverse image of an interval under is an element of the sigma-algebra on , because (closed) intervals generate the Borel sigma algebra on the reals. Since is a closed interval, and, for every , ,
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Thus,
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Being the inverse image of a Borel set under a -measurable function , each set in the countable intersection is an element of . Since -algebras are, by definition, closed under countable intersections, this shows that is -measurable, and the integral is well-defined (and possibly infinite).
Step 2. We will first show that
The definition of and monotonicity of imply that , for every and every . By monotonicity (or, more precisely, its narrower version established in Remark 5; see also Remark 4) of the Lebesgue integral,
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and
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Note that the limit on the right exists (finite or infinite) because, due to monotonicity (see Remark 5 and Remark 4), the sequence is non-decreasing.
End of Step 2.
We now prove the reverse inequality. We seek to show that
- .
Proof using Fatou's lemma. Per Remark 3, the inequality we want to prove is equivalent to
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But the latter follows immediately from Fatou's lemma, and the proof is complete.
Independent proof. To prove the inequality without using Fatou's lemma, we need some extra machinery. Denote by the set of simple -measurable functions such that on .
Step 3. Given a simple function and a real number , define
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Then , , and .
Step 3a. To prove the first claim, let , for some finite collection of pairwise disjoint measurable sets such that , some (finite) non-negative constants , and denoting the indicator function of the set .
For every holds if and only if Given that the sets are pairwise disjoint,
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Since the pre-image of the Borel set under the measurable function is measurable, and -algebras, by definition, are closed under finite intersection and unions, the first claim follows.
Step 3b. To prove the second claim, note that, for each and every ,
Step 3c. To prove the third claim, we show that .
Indeed, if, to the contrary, , then an element
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exists such that , for every . Taking the limit as , we get
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But by initial assumption, . This is a contradiction.
Step 4. For every simple -measurable non-negative function ,
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To prove this, define . By Lemma 1, is a measure on . By "continuity from below" (Lemma 2),
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as required.
Step 5. We now prove that, for every ,
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Indeed, using the definition of , the non-negativity of , and the monotonicity of the Lebesgue integral (see Remark 5 and Remark 4), we have
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for every . In accordance with Step 4, as , the inequality becomes
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Taking the limit as yields
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as required.
Step 6. We are now able to prove the reverse inequality, i.e.
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Indeed, by non-negativity, and For the calculation below, the non-negativity of is essential. Applying the definition of the Lebesgue integral and the inequality established in Step 5, we have
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The proof is complete.
Relaxing the monotonicity assumption
Under similar hypotheses to Beppo Levi's theorem, it is possible to relax the hypothesis of monotonicity. As before, let be a measure space and . Again, will be a sequence of -measurable non-negative functions . However, we do not assume they are pointwise non-decreasing. Instead, we assume that converges for almost every , we define to be the pointwise limit of , and we assume additionally that pointwise almost everywhere for all . Then is -measurable, and exists, and
As before, measurability follows from the fact that almost everywhere. The interchange of limits and integrals is then an easy consequence of Fatou's lemma. One has
by Fatou's lemma, and then, by standard properties of limits and monotonicity, Therefore , and both are equal to . It follows that exists and equals .