Monotone convergence theorem – Wikipedia
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 convergence of monotonic sequences (sequences that are non-decreasing or non-increasing) that are also bounded. Informally, the theorems state that if a sequence is increasing and bounded above by a supremum, then the sequence will converge to the supremum; in the same way, if a sequence is decreasing and is bounded below by an infimum, it will converge to the infimum.
Convergence of a monotone sequence of real numbers[edit]
Lemma 1[edit]
If a sequence of real numbers is increasing and bounded above, then its supremum is the limit.
Proof[edit]
Let
(an)n∈N{displaystyle (a_{n})_{nin mathbb {N} }}{an}{displaystyle {a_{n}}} be such a sequence, and let
be the set of terms of
{an}{displaystyle {a_{n}}} . By assumption,
c=supn{an}{textstyle c=sup _{n}{a_{n}}} is non-empty and bounded above. By the least-upper-bound property of real numbers,
ε>0{displaystyle varepsilon >0} exists and is finite. Now, for every
N{displaystyle N}
aN>c−ε{displaystyle a_{N}>c-varepsilon } such that
c−ε{displaystyle c-varepsilon }
{an}{displaystyle {a_{n}}} is an upper bound of
c{displaystyle c} , which contradicts the definition of
(an)n∈N{displaystyle (a_{n})_{nin mathbb {N} }} . Then since
c{displaystyle c} is increasing, and
n>N{displaystyle n>N} is its upper bound, for every
|c−an|≤|c−aN|<ε{displaystyle |c-a_{n}|leq |c-a_{N}|
(an)n∈N{displaystyle (a_{n})_{nin mathbb {N} }} . Hence, by definition, the limit of
supn{an}.{textstyle sup _{n}{a_{n}}.} is
Lemma 2[edit]
If a sequence of real numbers is decreasing and bounded below, then its infimum is the limit.
Proof[edit]
The proof is similar to the proof for the case when the sequence is increasing and bounded above,
Theorem[edit]
If
(an)n∈N{displaystyle (a_{n})_{nin mathbb {N} }}is a monotone sequence of real numbers (i.e., if an ≤ an+1 for every n ≥ 1 or an ≥ an+1 for every n ≥ 1), then this sequence has a finite limit if and only if the sequence is bounded.[1]
Proof[edit]
- “If”-direction: The proof follows directly from the lemmas.
- “Only If”-direction: By (ε, δ)-definition of limit, every sequence
(an)n∈N{displaystyle (a_{n})_{nin mathbb {N} }} with a finite limit L{displaystyle L} is necessarily bounded.
Convergence of a monotone series[edit]
Theorem[edit]
If for all natural numbers j and k, aj,k is a non-negative real number and aj,k ≤ aj+1,k, then[2]: 168
- limj→∞∑kaj,k=∑klimj→∞aj,k.{displaystyle lim _{jto infty }sum _{k}a_{j,k}=sum _{k}lim _{jto infty }a_{j,k}.}
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 series 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). 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
-
- (1+1n)n=∑k=0n(nk)1nk=∑k=0n1k!×nn×n−1n×⋯×n−k+1n,{displaystyle left(1+{frac {1}{n}}right)^{n}=sum _{k=0}^{n}{binom {n}{k}}{frac {1}{n^{k}}}=sum _{k=0}^{n}{frac {1}{k!}}times {frac {n}{n}}times {frac {n-1}{n}}times cdots times {frac {n-k+1}{n}},}
where n approaches infinity (the limit of this series is e). Here the matrix entry in row n and column k is
- (nk)1nk=1k!×nn×n−1n×⋯×n−k+1n;{displaystyle {binom {n}{k}}{frac {1}{n^{k}}}={frac {1}{k!}}times {frac {n}{n}}times {frac {n-1}{n}}times cdots times {frac {n-k+1}{n}};}
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
(1+1/n)n{displaystyle (1+1/n)^{n}}1k!{displaystyle {frac {1}{k!}}} by taking the sum of the column limits, namely
.
Beppo Levi’s lemma[edit]
The following result is due to Beppo Levi, who proved a slight generalization in 1906 of an earlier result by Henri Lebesgue.[3] In what follows,
BR≥0{displaystyle operatorname {mathcal {B}} _{mathbb {R} _{geq 0}}}σ{displaystyle sigma } denotes the
[0,+∞]{displaystyle [0,+infty ]} -algebra of Borel sets on
BR≥0{displaystyle operatorname {mathcal {B}} _{mathbb {R} _{geq 0}}} . By definition,
{+∞}{displaystyle {+infty }} contains the set
R≥0.{displaystyle mathbb {R} _{geq 0}.} and all Borel subsets of
Theorem[edit]
Let
(Ω,Σ,μ){displaystyle (Omega ,Sigma ,mu )}X∈Σ{displaystyle Xin Sigma } be a measure space, and
{fk}k=1∞{displaystyle {f_{k}}_{k=1}^{infty }} . Consider a pointwise non-decreasing sequence
(Σ,BR≥0){displaystyle (Sigma ,operatorname {mathcal {B}} _{mathbb {R} _{geq 0}})} of
fk:X→[0,+∞]{displaystyle f_{k}:Xto [0,+infty ]} -measurable non-negative functions
k≥1{displaystyle {kgeq 1}} , i.e., for every
x∈X{displaystyle {xin X}} and every
,
- 0≤fk(x)≤fk+1(x)≤∞.{displaystyle 0leq f_{k}(x)leq f_{k+1}(x)leq infty .}
Set the pointwise limit of the sequence
{fn}{displaystyle {f_{n}}}f{displaystyle f} to be
x∈X{displaystyle xin X} . That is, for every
,
- f(x):=limk→∞fk(x).{displaystyle f(x):=lim _{kto infty }f_{k}(x).}
Then
f{displaystyle f}(Σ,BR≥0){displaystyle (Sigma ,operatorname {mathcal {B}} _{mathbb {R} _{geq 0}})} is
-measurable and
- limk→∞∫Xfkdμ=∫Xfdμ.{displaystyle lim _{kto infty }int _{X}f_{k},dmu =int _{X}f,dmu .}
Remark 1. The integrals may be finite or infinite.
Remark 2. The theorem remains true if its assumptions hold
μ{displaystyle mu }N{displaystyle N} -almost everywhere. In other words, it is enough that there is a null set
{fn(x)}{displaystyle {f_{n}(x)}} such that the sequence
x∈X∖N.{displaystyle {xin Xsetminus N}.} non-decreases for every
{fn}{displaystyle {f_{n}}} To see why this is true, we start with an observation that allowing the sequence
f{displaystyle f} to pointwise non-decrease almost everywhere causes its pointwise limit
N{displaystyle N} to be undefined on some null set
f{displaystyle f} . On that null set,
μ(N)=0,{displaystyle {mu (N)=0},} 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
k,{displaystyle k,} we have, for every
- ∫Xfkdμ=∫X∖Nfkdμ{displaystyle int _{X}f_{k},dmu =int _{Xsetminus N}f_{k},dmu } and ∫Xfdμ=∫X∖Nfdμ,{displaystyle int _{X}f,dmu =int _{Xsetminus N}f,dmu ,}
provided that
f{displaystyle f}(Σ,BR≥0){displaystyle (Sigma ,operatorname {mathcal {B}} _{mathbb {R} _{geq 0}})} is
: section 21.38 (These equalities follow directly from the definition of Lebesgue integral for a non-negative function).
-measurable.[4]Remark 3. Under assumptions of the theorem,
f(x)=lim infkfk(x)=lim supkfk(x)=supkfk(x){displaystyle textstyle f(x)=liminf _{k}f_{k}(x)=limsup _{k}f_{k}(x)=sup _{k}f_{k}(x)}
lim infk∫Xfkdμ=lim supk∫Xfkdμ=limk∫Xfkdμ=supk∫Xfkdμ{displaystyle textstyle liminf _{k}int _{X}f_{k},dmu =textstyle limsup _{k}int _{X}f_{k},dmu =lim _{k}int _{X}f_{k},dmu =sup _{k}int _{X}f_{k},dmu }
(Note that the second chain of equalities follows from Remark 5).
Remark 4. The proof below does not use any properties of 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 Lebesgue integral). In the proof below, we apply the monotonic property of Lebesgue integral to non-negative functions only. Specifically (see Remark 4), let the functions
f,g:X→[0,+∞]{displaystyle f,g:Xto [0,+infty ]}(Σ,BR≥0){displaystyle (Sigma ,operatorname {mathcal {B}} _{mathbb {R} _{geq 0}})} be
-measurable.
- If
f≤g{displaystyle fleq g} everywhere on X,{displaystyle X,} then
- ∫Xfdμ≤∫Xgdμ.{displaystyle int _{X}f,dmu leq int _{X}g,dmu .}
- If
X1,X2∈Σ{displaystyle X_{1},X_{2}in Sigma } and X1⊆X2,{displaystyle X_{1}subseteq X_{2},} then
- ∫X1fdμ≤∫X2fdμ.{displaystyle int _{X_{1}}f,dmu leq int _{X_{2}}f,dmu .}
Proof. Denote
SF(h){displaystyle operatorname {SF} (h)}(Σ,BR≥0){displaystyle (Sigma ,operatorname {mathcal {B}} _{mathbb {R} _{geq 0}})} the set of simple
s:X→[0,∞){displaystyle s:Xto [0,infty )} -measurable functions
X.{displaystyle X.} everywhere on
1. Since
f≤g,{displaystyle fleq g,}we have
- SF(f)⊆SF(g).{displaystyle operatorname {SF} (f)subseteq operatorname {SF} (g).}
By definition of Lebesgue integral and the properties of supremum,
- ∫Xfdμ=sups∈SF(f)∫Xsdμ≤sups∈SF(g)∫Xsdμ=∫Xgdμ.{displaystyle int _{X}f,dmu =sup _{sin {rm {SF}}(f)}int _{X}s,dmu leq sup _{sin {rm {SF}}(g)}int _{X}s,dmu =int _{X}g,dmu .}
2. Let
1X1{displaystyle {mathbf {1} }_{X_{1}}}X1.{displaystyle X_{1}.} be the indicator function of the set
It can be deduced from the definition of Lebesgue integral that
- ∫X2f⋅1X1dμ=∫X1fdμ{displaystyle int _{X_{2}}fcdot {mathbf {1} }_{X_{1}},dmu =int _{X_{1}}f,dmu }
if we notice that, for every
s∈SF(f⋅1X1),{displaystyle sin {rm {SF}}(fcdot {mathbf {1} }_{X_{1}}),}s=0{displaystyle s=0}
X1.{displaystyle X_{1}.} outside of
f⋅1X1≤f{displaystyle fcdot {mathbf {1} }_{X_{1}}leq f} Combined with the previous property, the inequality
implies
- ∫X1fdμ=∫X2f⋅1X1dμ≤∫X2fdμ.{displaystyle int _{X_{1}}f,dmu =int _{X_{2}}fcdot {mathbf {1} }_{X_{1}},dmu leq int _{X_{2}}f,dmu .}
Proof[edit]
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[edit]
Lebesgue integral as measure[edit]
Lemma 1. Let
(Ω,Σ,μ){displaystyle (Omega ,Sigma ,mu )}(Σ,BR≥0){displaystyle (Sigma ,operatorname {mathcal {B}} _{mathbb {R} _{geq 0}})} be a measurable space. Consider a simple
s:Ω→R≥0{displaystyle s:Omega to {mathbb {R} _{geq 0}}} -measurable non-negative function
S⊆Ω{displaystyle Ssubseteq Omega } . For a subset
, define
- ν(S)=∫Ssdμ.{displaystyle nu (S)=int _{S}s,dmu .}
Then
ν{displaystyle nu }Ω{displaystyle Omega } is a measure on
.
Proof[edit]
Monotonicity follows from Remark 5. Here, we will only prove countable additivity, leaving the rest up to the reader. Let
S=⋃i=1∞Si{displaystyle S=bigcup _{i=1}^{infty }S_{i}}Si{displaystyle S_{i}} , where all the sets
are pairwise disjoint. Due to simplicity,
- s=∑i=1nci⋅1Ai,{displaystyle s=sum _{i=1}^{n}c_{i}cdot {mathbf {1} }_{A_{i}},}
for some finite non-negative constants
ci∈R≥0{displaystyle c_{i}in {mathbb {R} }_{geq 0}}Ai∈Σ{displaystyle A_{i}in Sigma } and pairwise disjoint sets
⋃i=1nAi=Ω{displaystyle bigcup _{i=1}^{n}A_{i}=Omega } such that
. By definition of Lebesgue integral,
- ν(S)=∑i=1nci⋅μ(S∩Ai)=∑i=1nci⋅μ((⋃j=1∞Sj)∩Ai)=∑i=1nci⋅μ(⋃j=1∞(Sj∩Ai)){displaystyle {begin{aligned}nu (S)&=sum _{i=1}^{n}c_{i}cdot mu (Scap A_{i})\&=sum _{i=1}^{n}c_{i}cdot mu left(left(bigcup _{j=1}^{infty }S_{j}right)cap A_{i}right)\&=sum _{i=1}^{n}c_{i}cdot mu left(bigcup _{j=1}^{infty }(S_{j}cap A_{i})right)end{aligned}}}
Since all the sets
Sj∩Ai{displaystyle S_{j}cap A_{i}}μ{displaystyle mu } are pairwise disjoint, the countable additivity of
gives us
- ∑i=1nci⋅μ(⋃j=1∞(Sj∩Ai))=∑i=1nci⋅∑j=1∞μ(Sj∩Ai).{displaystyle sum _{i=1}^{n}c_{i}cdot mu left(bigcup _{j=1}^{infty }(S_{j}cap A_{i})right)=sum _{i=1}^{n}c_{i}cdot sum _{j=1}^{infty }mu (S_{j}cap A_{i}).}
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,
- ∑i=1nci⋅∑j=1∞μ(Sj∩Ai)=∑j=1∞∑i=1nci⋅μ(Sj∩Ai)=∑j=1∞∫Sjsdμ=∑j=1∞ν(Sj),{displaystyle {begin{aligned}sum _{i=1}^{n}c_{i}cdot sum _{j=1}^{infty }mu (S_{j}cap A_{i})&=sum _{j=1}^{infty }sum _{i=1}^{n}c_{i}cdot mu (S_{j}cap A_{i})\&=sum _{j=1}^{infty }int _{S_{j}}s,dmu \&=sum _{j=1}^{infty }nu (S_{j}),end{aligned}}}
as required.
“Continuity from below”[edit]
The following property is a direct consequence of the definition of measure.
Lemma 2. Let
μ{displaystyle mu }S=⋃i=1∞Si{displaystyle S=bigcup _{i=1}^{infty }S_{i}} be a measure, and
, where
- S1⊆⋯⊆Si⊆Si+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)=limiμ(Si).{displaystyle mu (S)=lim _{i}mu (S_{i}).}
Proof of theorem[edit]
Step 1. We begin by showing that
f{displaystyle f}(Σ,BR≥0){displaystyle (Sigma ,operatorname {mathcal {B}} _{mathbb {R} _{geq 0}})} is
: section 21.3
–measurable.[4]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
[0,t]{displaystyle [0,t]}f{displaystyle f} under
Σ{displaystyle Sigma } is an element of the sigma-algebra
X{displaystyle X} on
[0,t]{displaystyle [0,t]} , because (closed) intervals generate the Borel sigma algebra on the reals. Since
k{displaystyle k} is a closed interval, and, for every
0≤fk(x)≤f(x){displaystyle 0leq f_{k}(x)leq f(x)} ,
,
- 0≤f(x)≤t⇔[∀k0≤fk(x)≤t].{displaystyle 0leq f(x)leq tquad Leftrightarrow quad {Bigl [}forall kquad 0leq f_{k}(x)leq t{Bigr ]}.}
Thus,
- {x∈X∣0≤f(x)≤t}=⋂k{x∈X∣0≤fk(x)≤t}.{displaystyle {xin Xmid 0leq f(x)leq t}=bigcap _{k}{xin Xmid 0leq f_{k}(x)leq t}.}
Being the inverse image of a Borel set under a
(Σ,BR≥0){displaystyle (Sigma ,operatorname {mathcal {B}} _{mathbb {R} _{geq 0}})}fk{displaystyle f_{k}} -measurable function
Σ{displaystyle Sigma } , each set in the countable intersection is an element of
σ{displaystyle sigma } . Since
f{displaystyle f} -algebras are, by definition, closed under countable intersections, this shows that
(Σ,BR≥0){displaystyle (Sigma ,operatorname {mathcal {B}} _{mathbb {R} _{geq 0}})} is
∫Xfdμ{displaystyle textstyle int _{X}f,dmu } -measurable, and the integral
is well defined (and possibly infinite).
Step 2. We will first show that
∫Xfdμ≥limk∫Xfkdμ.{displaystyle textstyle int _{X}f,dmu geq lim _{k}int _{X}f_{k},dmu .}
The definition of
f{displaystyle f}{fk}{displaystyle {f_{k}}} and monotonicity of
f(x)≥fk(x){displaystyle f(x)geq f_{k}(x)} imply that
k{displaystyle k} , for every
x∈X{displaystyle xin X} and every
. By monotonicity (or, more precisely, its narrower version established in Remark 5; see also Remark 4) of Lebesgue integral,
- ∫Xfdμ≥∫Xfkdμ,{displaystyle int _{X}f,dmu geq int _{X}f_{k},dmu ,}
and
- ∫Xfdμ≥limk∫Xfkdμ.{displaystyle int _{X}f,dmu geq lim _{k}int _{X}f_{k},dmu .}
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
- ∫Xfdμ≤limk∫Xfkdμ{displaystyle int _{X}f,dmu leq lim _{k}int _{X}f_{k},dmu } .
Proof using Fatou’s lemma. Per Remark 3, the inequality we want to prove is equivalent to
- ∫Xlim infkfk(x)dμ≤lim infk∫Xfkdμ.{displaystyle int _{X}liminf _{k}f_{k}(x),dmu leq liminf _{k}int _{X}f_{k},dmu .}
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
SF(f){displaystyle operatorname {SF} (f)}(Σ,BR≥0){displaystyle (Sigma ,operatorname {mathcal {B}} _{mathbb {R} _{geq 0}})} the set of simple
s:X→[0,∞){displaystyle s:Xto [0,infty )} -measurable functions
X{displaystyle X} on
.
Step 3. Given a simple function
s∈SF(f){displaystyle sin operatorname {SF} (f)}t∈(0,1){displaystyle tin (0,1)} and a real number
, define
- Bks,t={x∈X∣t⋅s(x)≤fk(x)}⊆X.{displaystyle B_{k}^{s,t}={xin Xmid tcdot s(x)leq f_{k}(x)}subseteq X.}
Then
Bks,t∈Σ{displaystyle B_{k}^{s,t}in Sigma }Bks,t⊆Bk+1s,t{displaystyle B_{k}^{s,t}subseteq B_{k+1}^{s,t}} ,
X=⋃kBks,t{displaystyle textstyle X=bigcup _{k}B_{k}^{s,t}} , and
.
Step 3a. To prove the first claim, let
s=∑i=1mci⋅1Ai{displaystyle textstyle s=sum _{i=1}^{m}c_{i}cdot {mathbf {1} }_{A_{i}}}Ai∈Σ{displaystyle A_{i}in Sigma } , for some finite collection of pairwise disjoint measurable sets
X=∪i=1mAi{displaystyle textstyle X=cup _{i=1}^{m}A_{i}} such that
ci∈R≥0{displaystyle c_{i}in {mathbb {R} }_{geq 0}} , some (finite) non-negative constants
1Ai{displaystyle {mathbf {1} }_{A_{i}}} , and
Ai{displaystyle A_{i}} denoting the indicator function of the set
.
For every
x∈Ai,{displaystyle xin A_{i},}t⋅s(x)≤fk(x){displaystyle tcdot s(x)leq f_{k}(x)}
fk(x)∈[t⋅ci,+∞].{displaystyle f_{k}(x)in [tcdot c_{i},+infty ].} holds if and only if
Ai{displaystyle A_{i}} Given that the sets
are pairwise disjoint,
- Bks,t=⋃i=1m(fk−1([t⋅ci,+∞])∩Ai).{displaystyle B_{k}^{s,t}=bigcup _{i=1}^{m}{Bigl (}f_{k}^{-1}{Bigl (}[tcdot c_{i},+infty ]{Bigr )}cap A_{i}{Bigr )}.}
Since the pre-image
fk−1([t⋅ci,+∞]){displaystyle f_{k}^{-1}{Bigl (}[tcdot c_{i},+infty ]{Bigr )}}
fk{displaystyle f_{k}} under the measurable function
σ{displaystyle sigma } 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
k{displaystyle k}x∈X{displaystyle xin X} and every
fk(x)≤fk+1(x).{displaystyle f_{k}(x)leq f_{k+1}(x).} ,
Step 3c. To prove the third claim, we show that
X⊆⋃kBks,t{displaystyle textstyle Xsubseteq bigcup _{k}B_{k}^{s,t}}.
Indeed, if, to the contrary,
X⊈⋃kBks,t{displaystyle textstyle Xnot subseteq bigcup _{k}B_{k}^{s,t}}, then an element
- x0∈X∖⋃kBks,t=⋂k(X∖Bks,t){displaystyle textstyle x_{0}in Xsetminus bigcup _{k}B_{k}^{s,t}=bigcap _{k}(Xsetminus B_{k}^{s,t})}
exists such that
fk(x0)<t⋅s(x0){displaystyle f_{k}(x_{0})k{displaystyle k} , for every
k→∞{displaystyle kto infty } . Taking the limit as
, we get
-
f(x0)≤t⋅s(x0)<s(x0).{displaystyle f(x_{0})leq tcdot s(x_{0})
But by initial assumption,
s≤f{displaystyle sleq f}. This is a contradiction.
Step 4. For every simple
(Σ,BR≥0){displaystyle (Sigma ,operatorname {mathcal {B}} _{mathbb {R} _{geq 0}})}s2{displaystyle s_{2}} -measurable non-negative function
,
- limn∫Bns,ts2dμ=∫Xs2dμ.{displaystyle lim _{n}int _{B_{n}^{s,t}}s_{2},dmu =int _{X}s_{2},dmu .}
To prove this, define
ν(S)=∫Ss2dμ{displaystyle textstyle nu (S)=int _{S}s_{2},dmu }ν(S){displaystyle nu (S)} . By Lemma 1,
Ω{displaystyle Omega } is a measure on
. By “continuity from below” (Lemma 2),
- limn∫Bns,ts2dμ=limnν(Bns,t)=ν(X)=∫Xs2dμ,{displaystyle lim _{n}int _{B_{n}^{s,t}}s_{2},dmu =lim _{n}nu (B_{n}^{s,t})=nu (X)=int _{X}s_{2},dmu ,}
as required.
Step 5. We now prove that, for every
s∈SF(f){displaystyle sin operatorname {SF} (f)},
- ∫Xsdμ≤limk∫Xfkdμ.{displaystyle int _{X}s,dmu leq lim _{k}int _{X}f_{k},dmu .}
Indeed, using the definition of
Bks,t{displaystyle B_{k}^{s,t}}fk{displaystyle f_{k}} , the non-negativity of
, and the monotonicity of Lebesgue integral (see Remark 5 and Remark 4), we have
- ∫Bks,tt⋅sdμ≤∫Bks,tfkdμ≤∫Xfkdμ,{displaystyle int _{B_{k}^{s,t}}tcdot s,dmu leq int _{B_{k}^{s,t}}f_{k},dmu leq int _{X}f_{k},dmu ,}
for every
k≥1{displaystyle kgeq 1}k→∞{displaystyle kto infty } . In accordance with Step 4, as
, the inequality becomes
- t∫Xsdμ≤limk∫Xfkdμ.{displaystyle tint _{X}s,dmu leq lim _{k}int _{X}f_{k},dmu .}
Taking the limit as
t↑1{displaystyle tuparrow 1}yields
- ∫Xsdμ≤limk∫Xfkdμ,{displaystyle int _{X}s,dmu leq lim _{k}int _{X}f_{k},dmu ,}
as required.
Step 6. We are now able to prove the reverse inequality, i.e.
- ∫Xfdμ≤limk∫Xfkdμ.{displaystyle int _{X}f,dmu leq lim _{k}int _{X}f_{k},dmu .}
Indeed, by non-negativity,
f+=f{displaystyle f_{+}=f}f−=0.{displaystyle f_{-}=0.} and
f{displaystyle f} For the calculation below, the non-negativity of
is essential. Applying the definition of Lebesgue integral and the inequality established in Step 5, we have
- ∫Xfdμ=sups∈SF(f)∫Xsdμ≤limk∫Xfkdμ.{displaystyle int _{X}f,dmu =sup _{sin operatorname {SF} (f)}int _{X}s,dmu leq lim _{k}int _{X}f_{k},dmu .}
The proof is complete.
See also[edit]
- ^ A generalisation of this theorem was given by Bibby, John (1974). “Axiomatisations of the average and a further generalisation of monotonic sequences”. Glasgow Mathematical Journal. 15 (1): 63–65. doi:10.1017/S0017089500002135.
- ^ See for instance Yeh, J. (2006). Real Analysis: Theory of Measure and Integration. Hackensack, NJ: World Scientific. ISBN 981-256-653-8.
- ^ Schappacher, Norbert; Schoof, René (1996), “Beppo Levi and the arithmetic of elliptic curves” (PDF), The Mathematical Intelligencer, 18 (1): 60, doi:10.1007/bf03024818, MR 1381581, S2CID 125072148, Zbl 0849.01036
- ^ a b See for instance Schechter, Erik (1997). Handbook of Analysis and Its Foundations. San Diego: Academic Press. ISBN 0-12-622760-8.
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