Monotone convergence theorem
Encyclopedia
In mathematics
, the monotone convergence theorem is any of there are several theorems. Some major examples are presented here.
of real number
s (e.g., if ak ≤ ak+1), then this sequence has a finite
limit if and only if the sequence is bounded.
is bounded above, then it is convergent and the limit is 
.
Since
is non-empty and by assumption, it is bounded above, therefore, by the Least upper bound property of real numbers, 
exists and is finite. Now for every 
, there exists 
such that 
, since otherwise 
is an upper bound of 
, which contradicts to 
being 
. Then since 
is increasing, 
, hence by definition, the limit of 
is 
is the limit.
The theorem states that if you have an infinite matrix of non-negative real numbers such that
then the limit of the sums of the rows is equal to the sum of the series whose term k is given by the limit of column k (which is also its supremum
). The series has a convergent sum if and only if the (weakly increasing) sequence of row sums is bounded and therefore convergent.
As an example, consider the infinite series of rows
where n approaches infinity (the limit of this series is e
). Here the matrix entry in row n and column k is
the columns (fixed k) are indeed weakly increasing with n and bounded (by 1/k!), while the rows only have finitely many nonzero terms, so condition 2 is satisfied; the theorem now says that you can compute the limit of the row sums
by taking the sum of the column limits, namely 
.
's theorem.
. Let
be a pointwise non-decreasing sequence of [0, ∞]-valued Σ–measurable
functions, i.e. for every k ≥ 1 and every x in X,
Next, set the pointwise limit of the sequence
to be f. That is, for every x in X,
Then f is Σ–measurable
(see for instance section 21.38) and
Remark. If the sequence
satisfies the assumptions μ–almost everywhere, one can find a set N ∈ Σ with μ(N) = 0 such that the sequence 
is non-decreasing for every 
. The result remains true because for every k,
. To do this, it is sufficient to show that the inverse image of an interval [0, t] under f is an element of the sigma algebra
Σ on X, because (closed) intervals generate the Borel sigma algebra on the reals. Let I = [0, t] be such a subinterval of [0, ∞]. Then
On the other hand, since I is a closed interval,
Thus,
Note that each set in the countable intersection is an element of Σ because it is the inverse image of a Borel subset under a Σ-measurable
function
. Since sigma algebras are, by definition, closed under countable intersections, this shows that f is Σ-measurable. In general, the supremum of any countable family of measurable functions is also measurable.
Now we will prove the rest of the monotone convergence theorem. The fact that f is Σ-measurable implies that the expression
is well defined.
We will start by showing that
By the definition of the Lebesgue integral
,
where SF is the set of Σ-measurable simple functions
on X. Since
at every x ∈ X, we have that
Hence, since the supremum of a subset cannot be larger than that of the whole set, we have that:
and the limit on the right exists, since the sequence is monotonic.
We now prove the inequality in the other direction (which also follows from Fatou's lemma
), that is we seek to show that
It follows from the definition of integral, that there is a non-decreasing sequence (gk) of non-negative simple functions such that gk ≤ f and such that
It suffices to prove that for each
,
because if this is true for each k, then the limit of the left-hand side will also be less than or equal to the right-hand side.
We will show that if gk is a simple function and
for every x, then
Since the integral is linear, we may break up the function
into its constant value parts, reducing to the case in which 
is the indicator function of an element B of the sigma algebra Σ. In this case, we assume that 
is a sequence of measurable functions whose supremum at every point of B is greater than or equal to one.
To prove this result, fix ε > 0 and define the sequence of measurable sets
By monotonicity of the integral, it follows that for any
,
By the assumption that
, any x in B will be in 
for sufficiently high values of n, and therefore
Thus, we have that
Using the monotonicity property of measures, we can continue the above equalities as follows:
Taking k → ∞, and using the fact that this is true for any positive ε, the result follows.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the monotone convergence theorem is any of there are several theorems. Some major examples are presented here.
Theorem
If ak is a monotone sequenceSequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s (e.g., if ak ≤ ak+1), then this sequence has a finite
Finite
Finite is the opposite of infinite. It may refer to:*Finite set, having a number of elements given by some natural number*Finite verb, being inflected for person and for tense...
limit if and only if the sequence is bounded.
Proof
We prove that if an increasing sequence is bounded above, then it is convergent and the limit is .Since
is non-empty and by assumption, it is bounded above, therefore, by the Least upper bound property of real numbers, exists and is finite. Now for every , there exists such that , since otherwise is an upper bound of , which contradicts to being . Then since is increasing, , hence by definition, the limit of is Remark
If a sequence of real numbers is decreasing and bounded below, then its infimumInfimum
In mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used...
is the limit.
Theorem
If for all natural numbers j and k, aj,k is a non-negative real number and aj,k ≤ aj+1,k, then (see for instance page 168)The theorem states that if you have an infinite matrix of non-negative real numbers such that
- the columns are weakly increasing and bounded, and
- for each row, the seriesSeries (mathematics)A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
whose terms are given by this row has a convergent sum,
then the limit of the sums of the rows is equal to the sum of the series whose term k is given by the limit of column k (which is also its supremum
Supremum
In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...
). The series has a convergent sum if and only if the (weakly increasing) sequence of row sums is bounded and therefore convergent.
As an example, consider the infinite series of rows
where n approaches infinity (the limit of this series is e
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...
). Here the matrix entry in row n and column k is
the columns (fixed k) are indeed weakly increasing with n and bounded (by 1/k!), while the rows only have finitely many nonzero terms, so condition 2 is satisfied; the theorem now says that you can compute the limit of the row sums
by taking the sum of the column limits, namely .Lebesgue's monotone convergence theorem
This theorem generalizes the previous one, and is probably the most important monotone convergence theorem. It is also known as Beppo LeviBeppo Levi
Beppo Levi was an Italian mathematician. He published high-level academic articles and books not only in mathematics, but also in physics, history, philosophy, and pedagogy. Levi was a member of the Bologna Academy of Sciences and of the Accademia dei Lincei.- Early years :Beppo Levi was born on...
's theorem.
Theorem
Let (X, Σ, μ) be a measure spaceMeasure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
. Let
be a pointwise non-decreasing sequence of [0, ∞]-valued Σ–measurableMeasurable function
In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
functions, i.e. for every k ≥ 1 and every x in X,
Next, set the pointwise limit of the sequence
to be f. That is, for every x in X,Then f is Σ–measurable
Measurable function
In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
(see for instance section 21.38) and
Remark. If the sequence
satisfies the assumptions μ–almost everywhere, one can find a set N ∈ Σ with μ(N) = 0 such that the sequence is non-decreasing for every . The result remains true because for every k,Proof
We will first show that f is Σ–measurableMeasurable function
In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
. To do this, it is sufficient to show that the inverse image of an interval [0, t] under f is an element of the sigma algebra
Sigma-algebra
In mathematics, a σ-algebra is a technical concept for a collection of sets satisfying certain properties. The main use of σ-algebras is in the definition of measures; specifically, the collection of sets over which a measure is defined is a σ-algebra...
Σ on X, because (closed) intervals generate the Borel sigma algebra on the reals. Let I = [0, t] be such a subinterval of [0, ∞]. Then
On the other hand, since I is a closed interval,
Thus,
Note that each set in the countable intersection is an element of Σ because it is the inverse image of a Borel subset under a Σ-measurable
Measurable function
In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
function
. Since sigma algebras are, by definition, closed under countable intersections, this shows that f is Σ-measurable. In general, the supremum of any countable family of measurable functions is also measurable.Now we will prove the rest of the monotone convergence theorem. The fact that f is Σ-measurable implies that the expression
is well defined.We will start by showing that
By the definition of the Lebesgue integral
Lebesgue integration
In mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...
,
where SF is the set of Σ-measurable simple functions
Simple function
In the mathematical field of real analysis, a simple function is a real-valued function over a subset of the real line which attains only a finite number of values...
on X. Since
at every x ∈ X, we have thatHence, since the supremum of a subset cannot be larger than that of the whole set, we have that:
and the limit on the right exists, since the sequence is monotonic.
We now prove the inequality in the other direction (which also follows from Fatou's lemma
Fatou's lemma
In mathematics, Fatou's lemma establishes an inequality relating the integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions...
), that is we seek to show that
It follows from the definition of integral, that there is a non-decreasing sequence (gk) of non-negative simple functions such that gk ≤ f and such that
It suffices to prove that for each
,because if this is true for each k, then the limit of the left-hand side will also be less than or equal to the right-hand side.
We will show that if gk is a simple function and
for every x, then
Since the integral is linear, we may break up the function
into its constant value parts, reducing to the case in which is the indicator function of an element B of the sigma algebra Σ. In this case, we assume that is a sequence of measurable functions whose supremum at every point of B is greater than or equal to one.To prove this result, fix ε > 0 and define the sequence of measurable sets
By monotonicity of the integral, it follows that for any
,By the assumption that
, any x in B will be in for sufficiently high values of n, and thereforeThus, we have that
Using the monotonicity property of measures, we can continue the above equalities as follows:
Taking k → ∞, and using the fact that this is true for any positive ε, the result follows.