- Source: Monotone convergence theorem
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
a
1
≤
a
2
≤
a
3
≤
.
.
.
≤
K
{\displaystyle a_{1}\leq a_{2}\leq a_{3}\leq ...\leq K}
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
0
≤
a
i
,
1
≤
a
i
,
2
≤
⋯
{\displaystyle 0\leq a_{i,1}\leq a_{i,2}\leq \cdots }
, 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
0
≤
f
1
(
x
)
≤
f
2
(
x
)
≤
⋯
{\displaystyle 0\leq f_{1}(x)\leq f_{2}(x)\leq \cdots }
, 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
Every bounded-above monotonically nondecreasing sequence of real numbers is convergent in the real numbers because the supremum exists and is a real number. The proposition does not apply to rational numbers because the supremum of a sequence of rational numbers may be irrational.
= Proposition
=(A) For a non-decreasing and bounded-above sequence of real numbers
a
1
≤
a
2
≤
a
3
≤
.
.
.
≤
K
<
∞
,
{\displaystyle a_{1}\leq a_{2}\leq a_{3}\leq ...\leq K<\infty ,}
the limit
lim
n
→
∞
a
n
{\displaystyle \lim _{n\to \infty }a_{n}}
exists and equals its supremum:
lim
n
→
∞
a
n
=
sup
n
a
n
≤
K
.
{\displaystyle \lim _{n\to \infty }a_{n}=\sup _{n}a_{n}\leq K.}
(B) For a non-increasing and bounded-below sequence of real numbers
a
1
≥
a
2
≥
a
3
≥
⋯
≥
L
>
−
∞
,
{\displaystyle a_{1}\geq a_{2}\geq a_{3}\geq \cdots \geq L>-\infty ,}
the limit
lim
n
→
∞
a
n
{\displaystyle \lim _{n\to \infty }a_{n}}
exists and equals its infimum:
lim
n
→
∞
a
n
=
inf
n
a
n
≥
L
{\displaystyle \lim _{n\to \infty }a_{n}=\inf _{n}a_{n}\geq L}
.
= Proof
=Let
{
a
n
}
n
∈
N
{\displaystyle \{a_{n}\}_{n\in \mathbb {N} }}
be the set of values of
(
a
n
)
n
∈
N
{\displaystyle (a_{n})_{n\in \mathbb {N} }}
. By assumption,
{
a
n
}
{\displaystyle \{a_{n}\}}
is non-empty and bounded above by
K
{\displaystyle K}
. By the least-upper-bound property of real numbers,
c
=
sup
n
{
a
n
}
{\textstyle c=\sup _{n}\{a_{n}\}}
exists and
c
≤
K
{\displaystyle c\leq K}
. Now, for every
ε
>
0
{\displaystyle \varepsilon >0}
, there exists
N
{\displaystyle N}
such that
c
≥
a
N
>
c
−
ε
{\displaystyle c\geq a_{N}>c-\varepsilon }
, since otherwise
c
−
ε
{\displaystyle c-\varepsilon }
is a strictly smaller upper bound of
{
a
n
}
{\displaystyle \{a_{n}\}}
, contradicting the definition of the supremum
c
{\displaystyle c}
. Then since
(
a
n
)
n
∈
N
{\displaystyle (a_{n})_{n\in \mathbb {N} }}
is non decreasing, and
c
{\displaystyle c}
is an upper bound, for every
n
>
N
{\displaystyle n>N}
, we have
|
c
−
a
n
|
=
c
−
a
n
≤
c
−
a
N
=
|
c
−
a
N
|
<
ε
.
{\displaystyle |c-a_{n}|=c-a_{n}\leq c-a_{N}=|c-a_{N}|<\varepsilon .}
Hence, by definition
lim
n
→
∞
a
n
=
c
=
sup
n
a
n
{\displaystyle \lim _{n\to \infty }a_{n}=c=\sup _{n}a_{n}}
.
The proof of the (B) part is analogous or follows from (A) by considering
{
−
a
n
}
n
∈
N
{\displaystyle \{-a_{n}\}_{n\in \mathbb {N} }}
.
= Theorem
=If
(
a
n
)
n
∈
N
{\displaystyle (a_{n})_{n\in \mathbb {N} }}
is a monotone sequence of real numbers, i.e., if
a
n
≤
a
n
+
1
{\displaystyle a_{n}\leq a_{n+1}}
for every
n
≥
1
{\displaystyle n\geq 1}
or
a
n
≥
a
n
+
1
{\displaystyle a_{n}\geq a_{n+1}}
for every
n
≥
1
{\displaystyle n\geq 1}
, then this sequence has a finite limit if and only if the sequence is bounded.
= Proof
="If"-direction: The proof follows directly from the proposition.
"Only If"-direction: By (ε, δ)-definition of limit, every sequence
(
a
n
)
n
∈
N
{\displaystyle (a_{n})_{n\in \mathbb {N} }}
with a finite limit
L
{\displaystyle L}
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
∞
{\displaystyle \infty }
and
−
∞
{\displaystyle -\infty }
added.
R
¯
=
R
∪
{
−
∞
,
∞
}
{\displaystyle {\bar {\mathbb {R} }}=\mathbb {R} \cup \{-\infty ,\infty \}}
In the extended real numbers every set has a supremum (resp. infimum) which of course may be
∞
{\displaystyle \infty }
(resp.
−
∞
{\displaystyle -\infty }
) if the set is unbounded. An important use of the extended reals is that any set of non negative numbers
a
i
≥
0
,
i
∈
I
{\displaystyle a_{i}\geq 0,i\in I}
has a well defined summation order independent sum
∑
i
∈
I
a
i
=
sup
J
⊂
I
,
|
J
|
<
∞
∑
j
∈
J
a
j
∈
R
¯
≥
0
{\displaystyle \sum _{i\in I}a_{i}=\sup _{J\subset I,\ |J|<\infty }\sum _{j\in J}a_{j}\in {\bar {\mathbb {R} }}_{\geq 0}}
where
R
¯
≥
0
=
[
0
,
∞
]
⊂
R
¯
{\displaystyle {\bar {\mathbb {R} }}_{\geq 0}=[0,\infty ]\subset {\bar {\mathbb {R} }}}
are the upper extended non negative real numbers. For a series of non negative numbers
∑
i
=
1
∞
a
i
=
lim
k
→
∞
∑
i
=
1
k
a
i
=
sup
k
∑
i
=
1
k
a
i
=
sup
J
⊂
N
,
|
J
|
<
∞
∑
j
∈
J
a
j
=
∑
i
∈
N
a
i
,
{\displaystyle \sum _{i=1}^{\infty }a_{i}=\lim _{k\to \infty }\sum _{i=1}^{k}a_{i}=\sup _{k}\sum _{i=1}^{k}a_{i}=\sup _{J\subset \mathbb {N} ,|J|<\infty }\sum _{j\in J}a_{j}=\sum _{i\in \mathbb {N} }a_{i},}
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.
Monotone convergence of non negative sums
Let
a
i
,
k
≥
0
{\displaystyle a_{i,k}\geq 0}
be a sequence of non-negative real numbers indexed by natural numbers
i
{\displaystyle i}
and
k
{\displaystyle k}
. Suppose that
a
i
,
k
≤
a
i
,
k
+
1
{\displaystyle a_{i,k}\leq a_{i,k+1}}
for all
i
,
k
{\displaystyle i,k}
. Then: 168
sup
k
∑
i
a
i
,
k
=
∑
i
sup
k
a
i
,
k
∈
R
¯
≥
0
.
{\displaystyle \sup _{k}\sum _{i}a_{i,k}=\sum _{i}\sup _{k}a_{i,k}\in {\bar {\mathbb {R} }}_{\geq 0}.}
= Proof
=Since
a
i
,
k
≤
sup
k
a
i
,
k
{\displaystyle a_{i,k}\leq \sup _{k}a_{i,k}}
we have
∑
i
a
i
,
k
≤
∑
i
sup
k
a
i
,
k
{\displaystyle \sum _{i}a_{i,k}\leq \sum _{i}\sup _{k}a_{i,k}}
so
sup
k
∑
i
a
i
,
k
≤
∑
i
sup
k
a
i
,
k
{\displaystyle \sup _{k}\sum _{i}a_{i,k}\leq \sum _{i}\sup _{k}a_{i,k}}
.
Conversely, we can interchange sup and sum for finite sums by reverting to the limit definition, so
∑
i
=
1
N
sup
k
a
i
,
k
=
sup
k
∑
i
=
1
N
a
i
,
k
≤
sup
k
∑
i
=
1
∞
a
i
,
k
{\displaystyle \sum _{i=1}^{N}\sup _{k}a_{i,k}=\sup _{k}\sum _{i=1}^{N}a_{i,k}\leq \sup _{k}\sum _{i=1}^{\infty }a_{i,k}}
hence
∑
i
=
1
∞
sup
k
a
i
,
k
≤
sup
k
∑
i
=
1
∞
a
i
,
k
{\displaystyle \sum _{i=1}^{\infty }\sup _{k}a_{i,k}\leq \sup _{k}\sum _{i=1}^{\infty }a_{i,k}}
.
= Examples
=Matrices
The theorem states that if you have an infinite matrix of non-negative real numbers
a
i
,
k
≥
0
{\displaystyle a_{i,k}\geq 0}
such that the rows are weakly increasing and each is bounded
a
i
,
k
≤
K
i
{\displaystyle a_{i,k}\leq K_{i}}
where the bounds are summable
∑
i
K
i
<
∞
{\displaystyle \sum _{i}K_{i}<\infty }
then, for each column, the non decreasing column sums
∑
i
a
i
,
k
≤
∑
K
i
{\displaystyle \sum _{i}a_{i,k}\leq \sum K_{i}}
are bounded hence convergent, and the limit of the column sums is equal to the sum of the "limit column"
sup
k
a
i
,
k
{\displaystyle \sup _{k}a_{i,k}}
which element wise is the supremum over the row.
e
Consider the expansion
(
1
+
1
k
)
k
=
∑
i
=
0
k
(
k
i
)
1
k
i
{\displaystyle \left(1+{\frac {1}{k}}\right)^{k}=\sum _{i=0}^{k}{\binom {k}{i}}{\frac {1}{k^{i}}}}
Now set
a
i
,
k
=
(
k
i
)
1
k
i
=
1
i
!
⋅
k
k
⋅
k
−
1
k
⋅
⋯
k
−
i
+
1
k
.
{\displaystyle a_{i,k}={\binom {k}{i}}{\frac {1}{k^{i}}}={\frac {1}{i!}}\cdot {\frac {k}{k}}\cdot {\frac {k-1}{k}}\cdot \cdots {\frac {k-i+1}{k}}.}
for
i
≤
k
{\displaystyle i\leq k}
and
a
i
,
k
=
0
{\displaystyle a_{i,k}=0}
for
i
>
k
{\displaystyle i>k}
, then
0
≤
a
i
,
k
≤
a
i
,
k
+
1
{\displaystyle 0\leq a_{i,k}\leq a_{i,k+1}}
with
sup
k
a
i
,
k
=
1
i
!
<
∞
{\displaystyle \sup _{k}a_{i,k}={\frac {1}{i!}}<\infty }
and
(
1
+
1
k
)
k
=
∑
i
=
0
∞
a
i
,
k
{\displaystyle \left(1+{\frac {1}{k}}\right)^{k}=\sum _{i=0}^{\infty }a_{i,k}}
.
The right hand side is a non decreasing sequence in
k
{\displaystyle k}
, therefore
lim
k
→
∞
(
1
+
1
k
)
k
=
sup
k
∑
i
=
0
∞
a
i
,
k
=
∑
i
=
0
∞
sup
k
a
i
,
k
=
∑
i
=
0
∞
1
i
!
=
e
{\displaystyle \lim _{k\to \infty }\left(1+{\frac {1}{k}}\right)^{k}=\sup _{k}\sum _{i=0}^{\infty }a_{i,k}=\sum _{i=0}^{\infty }\sup _{k}a_{i,k}=\sum _{i=0}^{\infty }{\frac {1}{i!}}=e}
.
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.
Let
B
R
¯
≥
0
{\displaystyle \operatorname {\mathcal {B}} _{{\bar {\mathbb {R} }}_{\geq 0}}}
denotes the
σ
{\displaystyle \sigma }
-algebra of Borel sets on the upper extended non negative real numbers
[
0
,
+
∞
]
{\displaystyle [0,+\infty ]}
. By definition,
B
R
¯
≥
0
{\displaystyle \operatorname {\mathcal {B}} _{{\bar {\mathbb {R} }}_{\geq 0}}}
contains the set
{
+
∞
}
{\displaystyle \{+\infty \}}
and all Borel subsets of
R
≥
0
.
{\displaystyle \mathbb {R} _{\geq 0}.}
= Theorem (monotone convergence theorem for non-negative measurable functions)
=Let
(
Ω
,
Σ
,
μ
)
{\displaystyle (\Omega ,\Sigma ,\mu )}
be a measure space, and
X
∈
Σ
{\displaystyle X\in \Sigma }
a measurable set. Let
{
f
k
}
k
=
1
∞
{\displaystyle \{f_{k}\}_{k=1}^{\infty }}
be a pointwise non-decreasing sequence of
(
Σ
,
B
R
¯
≥
0
)
{\displaystyle (\Sigma ,\operatorname {\mathcal {B}} _{{\bar {\mathbb {R} }}_{\geq 0}})}
-measurable non-negative functions, i.e. each function
f
k
:
X
→
[
0
,
+
∞
]
{\displaystyle f_{k}:X\to [0,+\infty ]}
is
(
Σ
,
B
R
¯
≥
0
)
{\displaystyle (\Sigma ,\operatorname {\mathcal {B}} _{{\bar {\mathbb {R} }}_{\geq 0}})}
-measurable and for every
k
≥
1
{\displaystyle {k\geq 1}}
and every
x
∈
X
{\displaystyle {x\in X}}
,
0
≤
…
≤
f
k
(
x
)
≤
f
k
+
1
(
x
)
≤
…
≤
∞
.
{\displaystyle 0\leq \ldots \leq f_{k}(x)\leq f_{k+1}(x)\leq \ldots \leq \infty .}
Then the pointwise supremum
sup
k
f
k
:
x
↦
sup
k
f
k
(
x
)
{\displaystyle \sup _{k}f_{k}:x\mapsto \sup _{k}f_{k}(x)}
is a
(
Σ
,
B
R
¯
≥
0
)
{\displaystyle (\Sigma ,\operatorname {\mathcal {B}} _{{\bar {\mathbb {R} }}_{\geq 0}})}
-measurable function and
sup
k
∫
X
f
k
d
μ
=
∫
X
sup
k
f
k
d
μ
.
{\displaystyle \sup _{k}\int _{X}f_{k}\,d\mu =\int _{X}\sup _{k}f_{k}\,d\mu .}
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
lim
k
→
∞
∫
X
f
k
(
x
)
d
μ
(
x
)
=
∫
X
lim
k
→
∞
f
k
(
x
)
d
μ
(
x
)
{\displaystyle \lim _{k\to \infty }\int _{X}f_{k}(x)\,d\mu (x)=\int _{X}\lim _{k\to \infty }f_{k}(x)\,d\mu (x)}
with the tacit understanding that the limits are allowed to be infinite.
Remark 3. The theorem remains true if its assumptions hold
μ
{\displaystyle \mu }
-almost everywhere. In other words, it is enough that there is a null set
N
{\displaystyle N}
such that the sequence
{
f
n
(
x
)
}
{\displaystyle \{f_{n}(x)\}}
non-decreases for every
x
∈
X
∖
N
.
{\displaystyle {x\in X\setminus N}.}
To see why this is true, we start with an observation that allowing the sequence
{
f
n
}
{\displaystyle \{f_{n}\}}
to pointwise non-decrease almost everywhere causes its pointwise limit
f
{\displaystyle f}
to be undefined on some null set
N
{\displaystyle N}
. On that null set,
f
{\displaystyle f}
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
μ
(
N
)
=
0
,
{\displaystyle {\mu (N)=0},}
we have, for every
k
,
{\displaystyle k,}
∫
X
f
k
d
μ
=
∫
X
∖
N
f
k
d
μ
{\displaystyle \int _{X}f_{k}\,d\mu =\int _{X\setminus N}f_{k}\,d\mu }
and
∫
X
f
d
μ
=
∫
X
∖
N
f
d
μ
,
{\displaystyle \int _{X}f\,d\mu =\int _{X\setminus N}f\,d\mu ,}
provided that
f
{\displaystyle f}
is
(
Σ
,
B
R
≥
0
)
{\displaystyle (\Sigma ,\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}})}
-measurable.: 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
f
,
g
:
X
→
[
0
,
+
∞
]
{\displaystyle f,g:X\to [0,+\infty ]}
be
(
Σ
,
B
R
¯
≥
0
)
{\displaystyle (\Sigma ,\operatorname {\mathcal {B}} _{{\bar {\mathbb {R} }}_{\geq 0}})}
-measurable.
If
f
≤
g
{\displaystyle f\leq g}
everywhere on
X
,
{\displaystyle X,}
then
∫
X
f
d
μ
≤
∫
X
g
d
μ
.
{\displaystyle \int _{X}f\,d\mu \leq \int _{X}g\,d\mu .}
If
X
1
,
X
2
∈
Σ
{\displaystyle X_{1},X_{2}\in \Sigma }
and
X
1
⊆
X
2
,
{\displaystyle X_{1}\subseteq X_{2},}
then
∫
X
1
f
d
μ
≤
∫
X
2
f
d
μ
.
{\displaystyle \int _{X_{1}}f\,d\mu \leq \int _{X_{2}}f\,d\mu .}
Proof. Denote by
SF
(
h
)
{\displaystyle \operatorname {SF} (h)}
the set of simple
(
Σ
,
B
R
≥
0
)
{\displaystyle (\Sigma ,\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}})}
-measurable functions
s
:
X
→
[
0
,
∞
)
{\displaystyle s:X\to [0,\infty )}
such that
0
≤
s
≤
h
{\displaystyle 0\leq s\leq h}
everywhere on
X
.
{\displaystyle X.}
1. Since
f
≤
g
,
{\displaystyle f\leq g,}
we have
SF
(
f
)
⊆
SF
(
g
)
,
{\displaystyle \operatorname {SF} (f)\subseteq \operatorname {SF} (g),}
hence
∫
X
f
d
μ
=
sup
s
∈
S
F
(
f
)
∫
X
s
d
μ
≤
sup
s
∈
S
F
(
g
)
∫
X
s
d
μ
=
∫
X
g
d
μ
.
{\displaystyle \int _{X}f\,d\mu =\sup _{s\in {\rm {SF}}(f)}\int _{X}s\,d\mu \leq \sup _{s\in {\rm {SF}}(g)}\int _{X}s\,d\mu =\int _{X}g\,d\mu .}
2. The functions
f
⋅
1
X
1
,
f
⋅
1
X
2
,
{\displaystyle f\cdot {\mathbf {1} }_{X_{1}},f\cdot {\mathbf {1} }_{X_{2}},}
where
1
X
i
{\displaystyle {\mathbf {1} }_{X_{i}}}
is the indicator function of
X
i
{\displaystyle X_{i}}
, are easily seen to be measurable and
f
⋅
1
X
1
≤
f
⋅
1
X
2
{\displaystyle f\cdot {\mathbf {1} }_{X_{1}}\leq f\cdot {\mathbf {1} }_{X_{2}}}
. Now apply 1.
= Lebesgue integral as measure =
Lemma 2. Let
(
Ω
,
Σ
,
μ
)
{\displaystyle (\Omega ,\Sigma ,\mu )}
be a measurable space. Consider a simple
(
Σ
,
B
R
≥
0
)
{\displaystyle (\Sigma ,\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}})}
-measurable non-negative function
s
:
Ω
→
R
≥
0
{\displaystyle s:\Omega \to {\mathbb {R} _{\geq 0}}}
. For a measurable subset
S
∈
Σ
{\displaystyle S\in \Sigma }
, define
ν
s
(
S
)
=
∫
S
s
d
μ
.
{\displaystyle \nu _{s}(S)=\int _{S}s\,d\mu .}
Then
ν
s
{\displaystyle \nu _{s}}
is a measure on
(
Ω
,
Σ
)
{\displaystyle (\Omega ,\Sigma )}
.
proof (lemma 2)
Write
s
=
∑
k
=
1
n
c
k
⋅
1
A
k
,
{\displaystyle s=\sum _{k=1}^{n}c_{k}\cdot {\mathbf {1} }_{A_{k}},}
with
c
k
∈
R
≥
0
{\displaystyle c_{k}\in {\mathbb {R} }_{\geq 0}}
and measurable sets
A
k
∈
Σ
{\displaystyle A_{k}\in \Sigma }
. Then
ν
s
(
S
)
=
∑
k
=
1
n
c
k
μ
(
S
∩
A
k
)
.
{\displaystyle \nu _{s}(S)=\sum _{k=1}^{n}c_{k}\mu (S\cap A_{k}).}
Since finite positive linear combinations of countably additive set functions are countably additive, to prove countable additivity of
ν
s
{\displaystyle \nu _{s}}
it suffices to prove that, the set function defined by
ν
A
(
S
)
=
μ
(
A
∩
S
)
{\displaystyle \nu _{A}(S)=\mu (A\cap S)}
is countably additive for all
A
∈
Σ
{\displaystyle A\in \Sigma }
. But this follows directly from the countable additivity of
μ
{\displaystyle \mu }
.
= Continuity from below =
Lemma 3. Let
μ
{\displaystyle \mu }
be a measure, and
S
=
⋃
i
=
1
∞
S
i
{\displaystyle S=\bigcup _{i=1}^{\infty }S_{i}}
, where
S
1
⊆
⋯
⊆
S
i
⊆
S
i
+
1
⊆
⋯
⊆
S
{\displaystyle S_{1}\subseteq \cdots \subseteq S_{i}\subseteq S_{i+1}\subseteq \cdots \subseteq S}
is a non-decreasing chain with all its sets
μ
{\displaystyle \mu }
-measurable. Then
μ
(
S
)
=
sup
i
μ
(
S
i
)
.
{\displaystyle \mu (S)=\sup _{i}\mu (S_{i}).}
proof (lemma 3)
Set
S
0
=
∅
{\displaystyle S_{0}=\emptyset }
, then
we decompose
S
=
∐
1
≤
i
S
i
∖
S
i
−
1
{\displaystyle S=\coprod _{1\leq i}S_{i}\setminus S_{i-1}}
as a countable disjoint union of measurable sets and likewise
S
k
=
∐
1
≤
i
≤
k
S
i
∖
S
i
−
1
{\displaystyle S_{k}=\coprod _{1\leq i\leq k}S_{i}\setminus S_{i-1}}
as a finite disjoint union. Therefore
μ
(
S
k
)
=
∑
i
=
1
k
μ
(
S
i
∖
S
i
−
1
)
{\displaystyle \mu (S_{k})=\sum _{i=1}^{k}\mu (S_{i}\setminus S_{i-1})}
, and
μ
(
S
)
=
∑
i
=
1
∞
μ
(
S
i
∖
S
i
−
1
)
{\displaystyle \mu (S)=\sum _{i=1}^{\infty }\mu (S_{i}\setminus S_{i-1})}
so
μ
(
S
)
=
sup
k
μ
(
S
k
)
{\displaystyle \mu (S)=\sup _{k}\mu (S_{k})}
.
Proof of theorem
Set
f
=
sup
k
f
k
{\displaystyle f=\sup _{k}f_{k}}
.
Denote by
SF
(
f
)
{\displaystyle \operatorname {SF} (f)}
the set of simple
(
Σ
,
B
R
≥
0
)
{\displaystyle (\Sigma ,\operatorname {\mathcal {B}} _{\mathbb {R} _{\geq 0}})}
-measurable functions
s
:
X
→
[
0
,
∞
)
{\displaystyle s:X\to [0,\infty )}
such that
0
≤
s
≤
f
{\displaystyle 0\leq s\leq f}
on
X
{\displaystyle X}
.
Step 1. The function
f
{\displaystyle f}
is
(
Σ
,
B
R
¯
≥
0
)
{\displaystyle (\Sigma ,\operatorname {\mathcal {B}} _{{\bar {\mathbb {R} }}_{\geq 0}})}
–measurable, and the integral
∫
X
f
d
μ
{\displaystyle \textstyle \int _{X}f\,d\mu }
is well-defined (albeit possibly infinite): section 21.3
From
0
≤
f
k
(
x
)
≤
∞
{\displaystyle 0\leq f_{k}(x)\leq \infty }
we get
0
≤
f
(
x
)
≤
∞
{\displaystyle 0\leq f(x)\leq \infty }
. Hence we have to show that
f
{\displaystyle f}
is
(
Σ
,
B
R
¯
≥
0
)
{\displaystyle (\Sigma ,\operatorname {\mathcal {B}} _{{\bar {\mathbb {R} }}_{\geq 0}})}
-measurable. To see this, it suffices to prove that
f
−
1
(
[
0
,
t
]
)
{\displaystyle f^{-1}([0,t])}
is
Σ
{\displaystyle \Sigma }
-measurable for all
0
≤
t
≤
∞
{\displaystyle 0\leq t\leq \infty }
, because the intervals
[
0
,
t
]
{\displaystyle [0,t]}
generate the Borel sigma algebra on the extended non negative reals
[
0
,
∞
]
{\displaystyle [0,\infty ]}
by complementing and taking countable intersections, complements and countable unions.
Now since the
f
k
(
x
)
{\displaystyle f_{k}(x)}
is a non decreasing sequence,
f
(
x
)
=
sup
k
f
k
(
x
)
≤
t
{\displaystyle f(x)=\sup _{k}f_{k}(x)\leq t}
if and only if
f
k
(
x
)
≤
t
{\displaystyle f_{k}(x)\leq t}
for all
k
{\displaystyle k}
. Since we already know that
f
≥
0
{\displaystyle f\geq 0}
and
f
k
≥
0
{\displaystyle f_{k}\geq 0}
we conclude that
f
−
1
(
[
0
,
t
]
)
=
⋂
k
f
k
−
1
(
[
0
,
t
]
)
.
{\displaystyle f^{-1}([0,t])=\bigcap _{k}f_{k}^{-1}([0,t]).}
Hence
f
−
1
(
[
0
,
t
]
)
{\displaystyle f^{-1}([0,t])}
is a measurable set,
being the countable intersection of the measurable sets
f
k
−
1
(
[
0
,
t
]
)
{\displaystyle f_{k}^{-1}([0,t])}
.
Since
f
≥
0
{\displaystyle f\geq 0}
the integral is well defined (but possibly infinite) as
∫
X
f
d
μ
=
sup
s
∈
S
F
(
f
)
∫
X
s
d
μ
{\displaystyle \int _{X}f\,d\mu =\sup _{s\in SF(f)}\int _{X}s\,d\mu }
.
Step 2. We have the inequality
sup
k
∫
X
f
k
d
μ
≤
∫
X
f
d
μ
{\displaystyle \sup _{k}\int _{X}f_{k}\,d\mu \leq \int _{X}f\,d\mu }
This is equivalent to
∫
X
f
k
(
x
)
d
μ
≤
∫
X
f
(
x
)
d
μ
{\displaystyle \int _{X}f_{k}(x)\,d\mu \leq \int _{X}f(x)\,d\mu }
for all
k
{\displaystyle k}
which follows directly from
f
k
(
x
)
≤
f
(
x
)
{\displaystyle f_{k}(x)\leq f(x)}
and "monotonicity of the integral" (lemma 1).
step 3 We have the reverse inequality
∫
X
f
d
μ
≤
sup
k
∫
X
f
k
d
μ
{\displaystyle \int _{X}f\,d\mu \leq \sup _{k}\int _{X}f_{k}\,d\mu }
.
By the definition of integral as a supremum step 3 is equivalent to
∫
X
s
d
μ
≤
sup
k
∫
X
f
k
d
μ
{\displaystyle \int _{X}s\,d\mu \leq \sup _{k}\int _{X}f_{k}\,d\mu }
for every
s
∈
SF
(
f
)
{\displaystyle s\in \operatorname {SF} (f)}
.
It is tempting to prove
∫
X
s
d
μ
≤
∫
X
f
k
d
μ
{\displaystyle \int _{X}s\,d\mu \leq \int _{X}f_{k}\,d\mu }
for
k
>
K
s
{\displaystyle k>K_{s}}
sufficiently large, but this does not work e.g. if
f
{\displaystyle f}
is itself simple and the
f
k
<
f
{\displaystyle f_{k}
. However, we can get ourself an "epsilon of room" to manoeuvre and avoid this problem.
Step 3 is also equivalent to
(
1
−
ε
)
∫
X
s
d
μ
=
∫
X
(
1
−
ε
)
s
d
μ
≤
sup
k
∫
X
f
k
d
μ
{\displaystyle (1-\varepsilon )\int _{X}s\,d\mu =\int _{X}(1-\varepsilon )s\,d\mu \leq \sup _{k}\int _{X}f_{k}\,d\mu }
for every simple function
s
∈
SF
(
f
)
{\displaystyle s\in \operatorname {SF} (f)}
and every
0
<
ε
≪
1
{\displaystyle 0<\varepsilon \ll 1}
where for the equality we used that the left hand side of the inequality is a finite sum. This we will prove.
Given
s
∈
SF
(
f
)
{\displaystyle s\in \operatorname {SF} (f)}
and
0
<
ε
≪
1
{\displaystyle 0<\varepsilon \ll 1}
, define
B
k
s
,
ε
=
{
x
∈
X
∣
(
1
−
ε
)
s
(
x
)
≤
f
k
(
x
)
}
⊆
X
.
{\displaystyle B_{k}^{s,\varepsilon }=\{x\in X\mid (1-\varepsilon )s(x)\leq f_{k}(x)\}\subseteq X.}
We claim the sets
B
k
s
,
ε
{\displaystyle B_{k}^{s,\varepsilon }}
have the following properties:
B
k
s
,
ε
{\displaystyle B_{k}^{s,\varepsilon }}
is
Σ
{\displaystyle \Sigma }
-measurable.
B
k
s
,
ε
⊆
B
k
+
1
s
,
ε
{\displaystyle B_{k}^{s,\varepsilon }\subseteq B_{k+1}^{s,\varepsilon }}
X
=
⋃
k
B
k
s
,
ε
{\displaystyle X=\bigcup _{k}B_{k}^{s,\varepsilon }}
Assuming the claim, by the definition of
B
k
s
,
ε
{\displaystyle B_{k}^{s,\varepsilon }}
and "monotonicity of the Lebesgue integral" (lemma 1) we have
∫
B
k
s
,
ε
(
1
−
ε
)
s
d
μ
≤
∫
B
k
s
,
ε
f
k
d
μ
≤
∫
X
f
k
d
μ
.
{\displaystyle \int _{B_{k}^{s,\varepsilon }}(1-\varepsilon )s\,d\mu \leq \int _{B_{k}^{s,\varepsilon }}f_{k}\,d\mu \leq \int _{X}f_{k}\,d\mu .}
Hence by "Lebesgue integral of a simple function as measure" (lemma 2), and "continuity from below" (lemma 3) we get:
sup
k
∫
B
k
s
,
ε
(
1
−
ε
)
s
d
μ
=
∫
X
(
1
−
ε
)
s
d
μ
≤
sup
k
∫
X
f
k
d
μ
.
{\displaystyle \sup _{k}\int _{B_{k}^{s,\varepsilon }}(1-\varepsilon )s\,d\mu =\int _{X}(1-\varepsilon )s\,d\mu \leq \sup _{k}\int _{X}f_{k}\,d\mu .}
which we set out to prove. Thus it remains to prove the claim.
Ad 1: Write
s
=
∑
1
≤
i
≤
m
c
i
⋅
1
A
i
{\displaystyle s=\sum _{1\leq i\leq m}c_{i}\cdot {\mathbf {1} }_{A_{i}}}
, for non-negative constants
c
i
∈
R
≥
0
{\displaystyle c_{i}\in \mathbb {R} _{\geq 0}}
, and measurable sets
A
i
∈
Σ
{\displaystyle A_{i}\in \Sigma }
, which we may assume are pairwise disjoint and with union
X
=
∐
i
=
1
m
A
i
{\displaystyle \textstyle X=\coprod _{i=1}^{m}A_{i}}
. Then for
x
∈
A
i
{\displaystyle x\in A_{i}}
we have
(
1
−
ε
)
s
(
x
)
≤
f
k
(
x
)
{\displaystyle (1-\varepsilon )s(x)\leq f_{k}(x)}
if and only if
f
k
(
x
)
∈
[
(
1
−
ε
)
c
i
,
∞
]
,
{\displaystyle f_{k}(x)\in [(1-\varepsilon )c_{i},\,\infty ],}
so
B
k
s
,
ε
=
∐
i
=
1
m
(
f
k
−
1
(
[
(
1
−
ε
)
c
i
,
∞
]
)
∩
A
i
)
{\displaystyle B_{k}^{s,\varepsilon }=\coprod _{i=1}^{m}{\Bigl (}f_{k}^{-1}{\Bigl (}[(1-\varepsilon )c_{i},\infty ]{\Bigr )}\cap A_{i}{\Bigr )}}
which is measurable since the
f
k
{\displaystyle f_{k}}
are measurable.
Ad 2: For
x
∈
B
k
s
,
ε
{\displaystyle x\in B_{k}^{s,\varepsilon }}
we have
(
1
−
ε
)
s
(
x
)
≤
f
k
(
x
)
≤
f
k
+
1
(
x
)
{\displaystyle (1-\varepsilon )s(x)\leq f_{k}(x)\leq f_{k+1}(x)}
so
x
∈
B
k
+
1
s
,
ε
.
{\displaystyle x\in B_{k+1}^{s,\varepsilon }.}
Ad 3: Fix
x
∈
X
{\displaystyle x\in X}
. If
s
(
x
)
=
0
{\displaystyle s(x)=0}
then
(
1
−
ε
)
s
(
x
)
=
0
≤
f
1
(
x
)
{\displaystyle (1-\varepsilon )s(x)=0\leq f_{1}(x)}
, hence
x
∈
B
1
s
,
ε
{\displaystyle x\in B_{1}^{s,\varepsilon }}
. Otherwise,
s
(
x
)
>
0
{\displaystyle s(x)>0}
and
(
1
−
ε
)
s
(
x
)
<
s
(
x
)
≤
f
(
x
)
=
sup
k
f
(
x
)
{\displaystyle (1-\varepsilon )s(x)
so
(
1
−
ε
)
s
(
x
)
<
f
N
x
(
x
)
{\displaystyle (1-\varepsilon )s(x)
for
N
x
{\displaystyle N_{x}}
sufficiently large, hence
x
∈
B
N
x
s
,
ε
{\displaystyle x\in B_{N_{x}}^{s,\varepsilon }}
.
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. As before, let
(
Ω
,
Σ
,
μ
)
{\displaystyle (\Omega ,\Sigma ,\mu )}
be a measure space and
X
∈
Σ
{\displaystyle X\in \Sigma }
. Again,
{
f
k
}
k
=
1
∞
{\displaystyle \{f_{k}\}_{k=1}^{\infty }}
will be a sequence of
(
Σ
,
B
R
≥
0
)
{\displaystyle (\Sigma ,{\mathcal {B}}_{\mathbb {R} _{\geq 0}})}
-measurable non-negative functions
f
k
:
X
→
[
0
,
+
∞
]
{\displaystyle f_{k}:X\to [0,+\infty ]}
. However, we do not assume they are pointwise non-decreasing. Instead, we assume that
{
f
k
(
x
)
}
k
=
1
∞
{\textstyle \{f_{k}(x)\}_{k=1}^{\infty }}
converges for almost every
x
{\displaystyle x}
, we define
f
{\displaystyle f}
to be the pointwise limit of
{
f
k
}
k
=
1
∞
{\displaystyle \{f_{k}\}_{k=1}^{\infty }}
, and we assume additionally that
f
k
≤
f
{\displaystyle f_{k}\leq f}
pointwise almost everywhere for all
k
{\displaystyle k}
. Then
f
{\displaystyle f}
is
(
Σ
,
B
R
≥
0
)
{\displaystyle (\Sigma ,{\mathcal {B}}_{\mathbb {R} _{\geq 0}})}
-measurable, and
lim
k
→
∞
∫
X
f
k
d
μ
{\textstyle \lim _{k\to \infty }\int _{X}f_{k}\,d\mu }
exists, and
lim
k
→
∞
∫
X
f
k
d
μ
=
∫
X
f
d
μ
.
{\displaystyle \lim _{k\to \infty }\int _{X}f_{k}\,d\mu =\int _{X}f\,d\mu .}
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
f
=
sup
k
f
k
=
lim
k
→
∞
f
k
=
lim inf
k
→
∞
f
k
{\textstyle f=\sup _{k}f_{k}=\lim _{k\to \infty }f_{k}=\liminf _{k\to \infty }f_{k}}
almost everywhere. The interchange of limits and integrals is then an easy consequence of Fatou's lemma. One has
∫
X
f
d
μ
=
∫
X
lim inf
k
f
k
d
μ
≤
lim inf
∫
X
f
k
d
μ
{\displaystyle \int _{X}f\,d\mu =\int _{X}\liminf _{k}f_{k}\,d\mu \leq \liminf \int _{X}f_{k}\,d\mu }
by Fatou's lemma, and then, since
∫
f
k
d
μ
≤
∫
f
k
+
1
d
μ
≤
∫
f
d
μ
{\displaystyle \int f_{k}\,d\mu \leq \int f_{k+1}\,d\mu \leq \int fd\mu }
(monotonicity),
lim inf
∫
X
f
k
d
μ
≤
lim sup
k
∫
X
f
k
d
μ
=
sup
k
∫
X
f
k
d
μ
≤
∫
X
f
d
μ
.
{\displaystyle \liminf \int _{X}f_{k}\,d\mu \leq \limsup _{k}\int _{X}f_{k}\,d\mu =\sup _{k}\int _{X}f_{k}\,d\mu \leq \int _{X}f\,d\mu .}
Therefore
∫
X
f
d
μ
=
lim inf
k
→
∞
∫
X
f
k
d
μ
=
lim sup
k
→
∞
∫
X
f
k
d
μ
=
lim
k
→
∞
∫
X
f
k
d
μ
=
sup
k
∫
X
f
k
d
μ
.
{\displaystyle \int _{X}f\,d\mu =\liminf _{k\to \infty }\int _{X}f_{k}\,d\mu =\limsup _{k\to \infty }\int _{X}f_{k}\,d\mu =\lim _{k\to \infty }\int _{X}f_{k}\,d\mu =\sup _{k}\int _{X}f_{k}\,d\mu .}
See also
Infinite series
Dominated convergence theorem
Notes
Kata Kunci Pencarian:
- Monotone convergence theorem
- Doob's martingale convergence theorems
- Monotone
- Bolzano–Weierstrass theorem
- Dominated convergence theorem
- Radon–Nikodym theorem
- Completeness of the real numbers
- Integral test for convergence
- Fatou's lemma
- Lebesgue integral
