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
Likewise, a non-increasing bounded-below sequence converges to its largest lower bound, its infimum.
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
(A) For a non-decreasing and bounded-above sequence of real numbers the limit
exists and equals its supremum: (B) For a non-increasing and bounded-below sequence of real numbers the limit
In the extended real numbers every set has a supremum (resp.
An important use of the extended reals is that any set of non negative numbers
In particular the sum of a series of non negative numbers does not depend on the order of summation.
The theorem states that if you have an infinite matrix of non-negative real numbers
, therefore 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.
-algebra of Borel sets on the upper extended non negative real numbers
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.
To see why this is true, we start with an observation that allowing the sequence
[5]: section 21.38 (These equalities follow directly from the definition of the Lebesgue integral for a non-negative function).
The theorem, thus, can be used to prove other basic properties, such as linearity, pertaining to Lebesgue integration.
Those not interested in this independency of the proof may skip the intermediate results below.
In the proof below, we apply the monotonic property of the Lebesgue integral to non-negative functions only.
as a countable disjoint union of measurable sets and likewise
generate the Borel sigma algebra on the extended non negative reals
However, we can get ourself an "epsilon of room" to manoeuvre and avoid this problem.
where for the equality we used that the left hand side of the inequality is a finite sum.
Under similar hypotheses to Beppo Levi's theorem, it is possible to relax the hypothesis of monotonicity.
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.
The interchange of limits and integrals is then an easy consequence of Fatou's lemma.