Theorems on the convergence of bounded monotonic sequences
In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the good convergence behaviour of monotonic sequences, i.e. sequences that are non-increasing, or non-decreasing. In its simplest form, it says that a non-decreasing bounded-above sequence of real numbers converges to its smallest upper bound, its supremum. Likewise, a non-increasing bounded-below sequence converges to its largest lower bound, its infimum. In particular, infinite sums of non-negative numbers converge to the supremum of the partial sums if and only if the partial sums are bounded.
For sums of non-negative increasing sequences , it says that taking the sum and the supremum can be interchanged.
In more advanced mathematics the monotone convergence theorem usually refers to a fundamental result in measure theory due to Lebesgue and Beppo Levi that says that for sequences of non-negative pointwise-increasing measurable functions, taking the integral and the supremum can be interchanged with the result being finite if either one is finite.
Convergence of a monotone sequence of real numbers
Bounded above monotone non decreasing sequences are convergent in the real numbers because suprema exist there. Note that the proposition is wrong when considered in the rational numbers.
Proposition
(A) For a non-decreasing and bounded-above sequence of real numbers
Let be the set of values of . By assumption, is non-empty and bounded above by . By the least-upper-bound property of real numbers, exists and . Now, for every , there exists such that , since otherwise is a strictly smaller upper bound of , contradicting the definition of the supremum . Then since is non decreasing, and is an upper bound, for every , we have
Hence, by definition .
The proof of the (B) part is analogous or follows from (A) by considering .
"If"-direction: The proof follows directly from the proposition.
"Only If"-direction: By (ε, δ)-definition of limit, every sequence with a finite limit is necessarily bounded.
Convergence of a monotone series
There is a variant of the proposition above where we allow unbounded sequences in the extended real numbers, the real numbers with and added.
In the extended real numbers every set has a supremum (resp. infimum) which of course may be (resp. ) if the set is unbounded. An important use of the extended reals is that any set of non negative numbers has a well defined summation order independent sum
where are the upper extended non negative real numbers. For a series of non negative numbers
so this sum coincides with the sum of a series if both are defined. In particular the sum of a series of non negative numbers does not depend on the order of summation.
Theorem (monotone convergence of non negative sums)
Let be a sequence of non-negative real numbers indexed by natural numbers and . Suppose that for all . Then[2]: 168
Remark
The suprema and the sums may be finite or infinite but the left hand side is finite if and only if the right hand side is.
proof: Since we have so . Conversely, we can interchange sup and sum for finite sums so
hence .
The theorem states that if you have an infinite matrix of non-negative real numbers such that
the rows are weakly increasing and each is bounded where the bounds are summable
then
for each column, the non decreasing column sums are bounded hence convergent, and the limit of the column sums is equal to the sum of the "limit column" which element wise is the supremum over the row.
As an example, consider the expansion
Now set
for and for , then with and
.
The right hand side is a non decreasing sequence in , therefore
.
Beppo Levi's lemma
The following result is a generalisation of the monotone convergence of non negative sums theorem above to the measure theoretic setting. It is a cornerstone of measure and integration theory with many applications and has Fatou's lemma and the dominated convergence theorem as direct consequence. It is due to Beppo Levi, who proved a slight generalization in 1906 of an earlier result by Henri Lebesgue.
[3][4]
Let denotes the -algebra of Borel sets on the upper extended non negative real numbers . By definition, contains the set and all Borel subsets of
Theorem (monotone convergence theorem for non-negative measurable functions)
Let be a measure space, and a measurable set. Let be a pointwise non-decreasing sequence of -measurable non-negative functions, i.e. each function is -measurable and for every and every ,
Then the pointwise supremum
is a -measurable function and
Remark 1. The integrals and the suprema may be finite or infinite, but the left-hand side is finite if and only if the right-hand side is.
Remark 2. Under the assumptions of the theorem,
(Note that the second chain of equalities follows from monoticity of the integral (lemma 2 below). Thus we can also write the conclusion of the theorem as
with the tacit understanding that the limits are allowed to be infinite.
Remark 3. 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.[5]: section 21.38 (These equalities follow directly from the definition of the Lebesgue integral for a non-negative function).
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.
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.
Intermediate results
We need three basic lemmas. In the proof below, we apply the monotonic property of the Lebesgue integral to non-negative functions only. Specifically (see Remark 4),
Monotonicity of the Lebesgue integral
lemma 1 . let the functions be -measurable.
If everywhere on then
If and then
Proof. Denote by the set of simple-measurable functions such that
everywhere on
1. Since we have
hence
2. The functions where is the indicator function of , are easily seen to be measurable and . Now apply 1.
Lebesgue integral as measure
Lemma 2. Let be a measurable space. Consider a simple -measurable non-negative function . For a measurable subset , define
Then is a measure on .
proof (lemma 2)
Write
with and measurable sets . Then
Since finite positive linear combinations of countably additive set functions are countably additive, to prove countable additivity of it suffices to prove that, the set function defined by is countably additive for all . But this follows directly from the countable additivity of .
Continuity from below
Lemma 3. Let be a measure, and , where
is a non-decreasing chain with all its sets -measurable. Then
proof (lemma 3)
Set , then
we decompose as a countable disjoint union of measurable sets and likewise as a finite disjoint union. Therefore
, and so .
Proof of theorem
Set .
Denote by the set of simple -measurable functions ( nor included!) such that
on .
Step 1. The function is –measurable, and the integral is well-defined (albeit possibly infinite)[5]: section 21.3
From we get . Hence we have to show that is -measurable. To see this, it suffices to prove that is -measurable for all , because the intervals generate the Borel sigma algebra on the extended non negative reals by complementing and taking countable intersections, complements and countable unions.
Now since the is a non decreasing sequence,
if and only if for all . Since we already know that and we conclude that
Hence is a measurable set,
being the countable intersection of the measurable sets .
Since the integral is well defined (but possibly infinite) as
.
Step 2. We have the inequality
This is equivalent to for all which follows directly from and "monotonicity of the integral" (lemma 1).
step 3 We have the reverse inequality
.
By the definition of integral as a supremum step 3 is equivalent to
for every .
It is tempting to prove for sufficiently large, but this does not work e.g. if is itself simple and the . However, we can get ourself an "epsilon of room" to manoeuvre and avoid this problem.
Step 3 is also equivalent to
for every simple function and every
where for the equality we used that the left hand side of the inequality is a finite sum. This we will prove.
Given and , define
We claim the sets have the following properties:
is -measurable.
Assuming the claim, by the definition of and "monotonicity of the Lebesgue integral" (lemma 1) we have
Hence by "Lebesgue integral of a simple function as measure" (lemma 2), and "continuity from below" (lemma 3) we get:
which we set out to prove. Thus it remains to prove the claim.
Ad 1: Write , for non-negative constants , and measurable sets , which we may assume are pairwise disjoint and with union . Then for we have if and only if so
which is measurable since the are measurable.
Ad 2: For we have so
Ad 3: Fix . If then , hence . Otherwise, and so for
sufficiently large, hence .
The proof of the monotone convergence theorem is complete.
Relaxing the monotonicity assumption
Under similar hypotheses to Beppo Levi's theorem, it is possible to relax the hypothesis of monotonicity.[6] 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
Proof based on Fatou's lemma
The proof can also be based on Fatou's lemma instead of a direct proof as above, because Fatou's lemma can be proved independent of the monotone convergence theorem.
However the monotone convergence theorem is in some ways more primitive than Fatou's lemma. It easily follows from the monotone convergence theorem and proof of Fatou's lemma is similar and arguably slightly less natural than the proof above.
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, since (monotonicity),
Therefore