Monotone convergence theorem: Difference between revisions
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In [[mathematics]], the |
In [[mathematics]], the'' 'monotone convergence Theorem''' are some of the few theorems. Examples of major expansion here. |
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== Convergence of a monotone sequence of real numbers == |
== Convergence of a monotone sequence of real numbers == |
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=== |
Theorem === === |
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If <math>\ |
If <math> \ {a_n \} </ math> is a monotone [[sequence]] of [[real number]] s (for example, if a'''''' <sub> n'' </ sub> ≤ a <sub>'''''''' n +1 </ sub> or a'''''''' <sub> n </ sub> ≥ <sub>'' a'''''' n +1 </ sub> for every n ≥ 1), then this sequence a [[block]] limit if and only if the line side is [[bounded sequence | survival]]. <ref> A generalization of this Theorem given by John Bibby (1974) "Axiomatisations the average and a further generalization of monotonic sequences," Glasgow mathematical Journal, vol. 15, pp. 63-65. </ Ref> |
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=== |
Proof === === |
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We prove that |
We prove that an increase <math> sequence \ {a_n \} </ math> is bounded above, then it is convergent and is opposed <math> \ sup \ limits_n \ {a_n \} </ math>. |
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Since <math>\ |
Since <math> \ {a_n \} </ math> is non-empty and the like, it is bounded above, then, from the [[at least Upper bound property]] by phone at real, <math> c = \ sup_n \ {a_n \} </ math> and government intervention. Now all <math> \ varepsilon> 0 </ math>, there exists <math> a_N </ math> such that <math> a_N> c - \ varepsilon </ math>, since otherwise < Math> c - \ varepsilon </ math> is a Upper bound of <math> \ {a_n \} </ math>, which contradicts the <math> c </ math> do <math> \ sup_n \ {a_n \} </ math>. Then since <math> \ {a_n \} </ math> is increased, <math> \ forall n> N, | c - a_n | = c - a_n \ leq c - a_N <\ varepsilon </ math>, hence by definition, the limit of <math> \ {a_n \} </ math> is <math> \ sup_n \ {a_n \}. </ Math> |
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=== |
Remark === === |
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If a sequence of real numbers |
If a sequence of real numbers and reduced survival below, then [[infimum]] is forbidden. |
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== Convergence of a monotone series == |
== Convergence of a monotone series == |
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=== |
Theorem === === |
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If |
If all the natural numbers'' j'' and'' k'','''''''' a <sub> j, k'''' </ sub> is a non- -not real number and a'''''''' <sub> j, k'''' </ sub> ≤'''''''' <sub> a j +1, '' k'' </ sub>, then (see for example <ref> {{Cite book | author = J Yeh | title = real test. theory of measure and integration | year = 2006} } </ ref> page 168) |
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:<math>\ |
: <math> \ Lim_ {j \ to \ infty} \ sum_k Ã_ {j, k} = \ sum_k \ lim_ {j \ to \ infty} Ã_ {j, k}. </ Mathematics> |
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The |
The Theorem that if you have an infinite matrix of non-negative real numbers such that |
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# |
# The line is weakly increasing and bounded, and |
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# |
# For each line, the [[series (mathematics) | series]] whose language be here together a convergent sum, |
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the limit of the sums of columns is equal to the sum of the series of time'' k'' is the limit of the box'' k'' (which is also his [[supremum]]) . The series has a convergent sum if and only if the (weakly increasing) sequence of line sums is bounded and therefore convergent. |
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As an example, consider the infinite series of |
As an example, consider the infinite series of line |
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<math> :: (1 +1 / n) ^ n = \ sum_ {k = 0} ^ {n} \ binom nk / n ^ k = \ sum_ {k = 0} ^ {n} \ {k frac1! } \ Times \ frac nn \ times \ frac {n-1} n \ times \ cdots \ times \ frac {nk +1} n </ math> |
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where |
where'' n'' approaches maximal (the limit of this series is [[e (mathematical constant) | e]]). Here is the matrix entry in line'' n'' and'' column'' is k |
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:<math>\ |
: <math> \ Binom nk / n ^ k = \ frac1 {k!} \ Time \ frac nn \ times \ frac {n-1} n \ times \ cdots \ times \ frac { nk +1} n; </ math> |
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the |
the line ('' security'' k) that is weakly increasing with'' n'' and bounded (by 1 /'''' k), when the line only finitely many nonzero terms, so pain is enough 2 for;'s Theorem now you include limits of line sums <math> (1 +1 / n) ^ n </ math> by taking the sum of the other restrictions, which <math> \ frac1 {k!} </ math>. |
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== Lebesgue's monotone convergence |
== Lebesgue's monotone convergence Theorem == |
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This |
This generalizes Theorem past one, and is probably the most important monotone convergence Theorem. It also called [[Beppo Levi]] 's Theorem. |
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=== |
Theorem === === |
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Let ('' |
Let ('''' X, Σ, μ'''') will have a [[measure (mathematics) | measurement location]]. Let <math> f_1, f_2, \ ldots </ math> be a pointwise non-Reduced sequence of [0, ∞]-valued Σ-[[Measurable Function | Measurable]] functions, eg '''' all'''' k ≥ 1 and all'' x'' in'' X'', |
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:<math> 0 \leq f_k(x) \leq f_{k+1}(x). \, </ |
: <math> 0 \ leq f_k (x) \ leq f_ {k +1} (x). \, </ Mathematics> |
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Next, |
Next, the pointwise limit of sequence <math> (f_ {n}) </ math> be'' f''. That is, for every'' x'' in'' X'', |
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:<math> |
: <math> F (x): = \ lim_ {k \ to \ infty} f_k (x). \, </ Mathematics> |
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Then |
Then'' f'' is Σ-[[Measurable Function | Measurable]] and |
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:<math>\ |
: <math> \ Lim_ {k \ to \ infty} \ int f_k \, \ mathrm {d} \ mu = \ int f \, \ mathrm {d} \ mu. </ Mathematics> |
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'''Remark.''' If the sequence <math>(f_k)</math> satisfies the |
'' 'Remark.''' If the sequence <math> (f_k) </ math> satisfies the estimate'''' μ-almost everywhere, one can find some'' '' N'' ∈ Σ with μ'' ('' N'') = 0 such that the sequence <math> (f_k (x)) </ math> is non-Reduce all <math> x \ notin N </ math>. The advantage is that because all'' k'', |
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:<math> \ |
: <math> \ Int f_k \, \ mathrm {d} \ mu = \ int_ {X \ backslash f_k N} \, \ mathrm {d} \ mu, \ \ text {and} \ \ int f \, \ mathrm {d} \ mu = \ int_ {X \ backslash N} f \, \ mathrm {d} \ mu </ math> |
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provided that f is Σ |
provided that f is Σ-Measurable (see example <ref name="SCHECHTER1997"> {{Cite book | author = Erik Schechter | title = Analysis and Its origins | year = 1997}} </ ref> section 21.38). |
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=== |
Proof === === |
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We |
We first show that the'' f'' is Σ-[[Measurable Function | Measurable]] (see Example <ref name="SCHECHTER1997"/> section 21.3). Do this, it is enough to show that the inverse image of an interval [0, t''''] in'' f'' is the element of [[Sigma-algebra | Sigma algebra]] Σ on'' X'', because the (closed) intervals to the [[Borel Sigma algebra]] for reals. I'''' = [0, t''''] be such a subinterval of [0, ∞]. Storage |
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:<math> |
: <math> F ^ {-1} (I) = \ {x \ in X \, | \, f (x) \ in I \}. </ Mathematics> |
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Since '' |
Since I'''' is a prolonged and close <math> \ forall k, f_k (x) \ Le f (x) </ math>, |
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:<math> |
: <math> F (x) \ in I \ Leftrightarrow f_k (x) \ in I, \ forall k \ in \ mathbb {N}. </ Mathematics> |
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Therefore, |
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Thus, |
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:<math>\{ |
: <math> \ {X \ in X \, | \, f (x) \ in I \} = \ bigcap_ {k \ in \ mathbb {N}} \ {x \ in X \, | \, f_k (x) \ in I \}. </ Math> |
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Note that each set in the countable intersection is |
Note that each set in the countable intersection is a element of Σ because it is the inverse image of a [[Borel subset]] in a Σ-[[Measurable Function | Measurable ]] Function <math> f_k </ math>. Since Sigma algebras is, by definition, closed in countable intersection, this indicates that'' f'' is Σ-Measurable. In general, the supremum of the countable family of functions is also Measurable Measurable. |
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Now we will prove the rest of the monotone convergence |
Now we will prove the rest of the monotone convergence Theorem. The fact that'' f'' is Σ-Measurable implies that information <math> \ int f \, \ mathrm {d} \ mu </ math> is well defined. |
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We |
We begin by showing that <math> \ int f \, \ mathrm {d} \ mu \ geq \ lim_k \ int f_k \, \ mathrm {d} \ mu. </ Mathematics> |
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By |
By means of the [[Lebesgue integration | Lebesgue integral]], |
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:<math> \ |
: <math> \ Int f \, \ mathrm {d} \ mu = \ sup \ {\ int g \, \ mathrm {d} \ mu \, | \, g \ in SF, \ g \ leq f \} </ math> |
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where SF is |
where SF is co-Measurable Σ [[simple Function | simple functions]] on'' X''. Since <math> f_k (x) \ leq f (x) </ math> of every'' x'' ∈'' X'', we have that |
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:<math>\ |
: <math> \ Left \ {\ int g \, \ mathrm {d} \ mu \, | \, g \ in SF, \ g \ leq f_k \ right \} \ subseteq \ left \ {\ int g \, \ mathrm {d} \ mu \, | \, g \ in SF, \ g \ leq f \ right \}. </ Math> |
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Hence, since the supremum of a subset |
Hence, since the supremum of a subset can not be serious than all set, we have that: |
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:<math> \ |
: <math> \ Int f \, \ mathrm {d} \ mu \ geq \ lim_k \ int f_k \, \ mathrm {d} \ mu </ math> |
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and |
and restrictions on political rights, since the sequence is monotonic. |
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We now prove the inequality in the other direction (which also follows from [[Fatou's |
We now prove the inequality in the other direction (which also follows from [[Fatou's Lemma]]), if we find that |
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:<math> \ |
: <math> \ Int f \, \ mathrm {d} \ mu \ leq \ lim_k \ int f_k \, \ mathrm {d} \ mu. </ Mathematics> |
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It follows from the definition of integral, |
It follows from the definition of the integral, with a non-Decreasing sequence (g <sub>'''''''' k </ sub>) of non-not easy eight functions such that'' g'' <sub>'' k'' </ sub> ≤'' f'' and such that |
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:<math> \ |
: <math> \ Lim_k \ int g_k \, \ mathrm {d} \ mu = \ int f \, \ mathrm {d} \ mu. </ Mathematics> |
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It suffices to prove that for |
It suffices to prove that for all <math> k \ in \ mathbb {N} </ math>, |
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:<math> |
: <math> \ Int g_k \, \ mathrm {d} \ mu \ leq \ lim_j \ int f_j \, \ mathrm {d} \ mu </ math> |
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because if this is true for |
because if this is true for all'' k'', then the limit of the left-hand side will be less than or equal to the right-hand side. |
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We will show that if '' |
We will show that if g'''''''' <sub> k </ sub> is a simple and Function |
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:<math> \ |
: <math> \ Lim_j f_j (x) \ geq g_k (x) \, </ math> |
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for every |
for every'' x'', then |
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:<math> \ |
: <math> \ Lim_j \ int f_j \, \ mathrm {d} \ mu \ geq \ int g_k \, \ mathrm {d} \ mu. </ Mathematics> |
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Since the integral is linear, we may |
Since the integral is linear, we may violate the Function <math> g_k </ math> in the constant value, reduce to the case in which <Math > g_k </ math> is the illustration of a Function element'' B'' of Sigma algebra Σ. In this case, we assume that <math> f_j </ math> is a sequence of functions which Measurable supremum of each point of'' B'' is more than or equal to a a. |
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By monotonicity of the integral, it follows that for any <math>n\in \mathbb{N}</math>, |
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By monotonicity of the integral, it says the following areas <math> n \ in \ mathbb {N} </ math>, |
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: <math> \ Mu \ left (\ bigcup_n B_n \ right) = \ lim_n \ mu (B_n) \ leq \ lim_n (1 - \ epsilon) ^ {-1} \ int f_n \, \ mathrm {d } \ mu. </ Mathematics> |
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==See also== |
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== |
See also == == |
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{{reflist}} |
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Note == == |
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[[Category:Articles containing proofs]] |
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{{Reflist}} |
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[[Category: |
[[Category: Articles evidence]] |
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[[Category: Theorems in the Calculus]] |
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Revision as of 11:13, 19 May 2013
In mathematics, the 'monotone convergence Theorem' are some of the few theorems. Examples of major expansion here.
Convergence of a monotone sequence of real numbers
Theorem === === If <math> \ {a_n \} </ math> is a monotone sequence of real number s (for example, if a' n </ sub> ≤ a ''' n +1 </ sub> or a''' n </ sub> ≥ a' n +1 </ sub> for every n ≥ 1), then this sequence a block limit if and only if the line side is survival. <ref> A generalization of this Theorem given by John Bibby (1974) "Axiomatisations the average and a further generalization of monotonic sequences," Glasgow mathematical Journal, vol. 15, pp. 63-65. </ Ref>
Proof === === We prove that an increase <math> sequence \ {a_n \} </ math> is bounded above, then it is convergent and is opposed <math> \ sup \ limits_n \ {a_n \} </ math>.
Since <math> \ {a_n \} </ math> is non-empty and the like, it is bounded above, then, from the at least Upper bound property by phone at real, <math> c = \ sup_n \ {a_n \} </ math> and government intervention. Now all <math> \ varepsilon> 0 </ math>, there exists <math> a_N </ math> such that <math> a_N> c - \ varepsilon </ math>, since otherwise < Math> c - \ varepsilon </ math> is a Upper bound of <math> \ {a_n \} </ math>, which contradicts the <math> c </ math> do <math> \ sup_n \ {a_n \} </ math>. Then since <math> \ {a_n \} </ math> is increased, <math> \ forall n> N, | c - a_n | = c - a_n \ leq c - a_N <\ varepsilon </ math>, hence by definition, the limit of <math> \ {a_n \} </ math> is <math> \ sup_n \ {a_n \}. </ Math>
Remark === === If a sequence of real numbers and reduced survival below, then infimum is forbidden.
Convergence of a monotone series
Theorem === === If all the natural numbers j and k,''' a j, k'' </ sub> is a non- -not real number and a''' j, k' </ sub> ≤''' a j +1, k </ sub>, then (see for example <ref> J Yeh (2006} } </ ref> page 168)
- <math> \ Lim_ {j \ to \ infty} \ sum_k Ã_ {j, k} = \ sum_k \ lim_ {j \ to \ infty} Ã_ {j, k}. </ Mathematics>
The Theorem that if you have an infinite matrix of non-negative real numbers such that
- The line is weakly increasing and bounded, and
- For each line, the series whose language be here together a convergent sum,
the limit of the sums of columns is equal to the sum of the series of time k is the limit of the box k (which is also his supremum) . The series has a convergent sum if and only if the (weakly increasing) sequence of line sums is bounded and therefore convergent.
As an example, consider the infinite series of line
<math> :: (1 +1 / n) ^ n = \ sum_ {k = 0} ^ {n} \ binom nk / n ^ k = \ sum_ {k = 0} ^ {n} \ {k frac1! } \ Times \ frac nn \ times \ frac {n-1} n \ times \ cdots \ times \ frac {nk +1} n </ math>
where n approaches maximal (the limit of this series is e). Here is the matrix entry in line n and column is k
- <math> \ Binom nk / n ^ k = \ frac1 {k!} \ Time \ frac nn \ times \ frac {n-1} n \ times \ cdots \ times \ frac { nk +1} n; </ math>
the line ( security k) that is weakly increasing with n and bounded (by 1 /' k), when the line only finitely many nonzero terms, so pain is enough 2 for;'s Theorem now you include limits of line sums <math> (1 +1 / n) ^ n </ math> by taking the sum of the other restrictions, which <math> \ frac1 {k!} </ math>.
Lebesgue's monotone convergence Theorem
This generalizes Theorem past one, and is probably the most important monotone convergence Theorem. It also called Beppo Levi 's Theorem.
Theorem === === Let (' X, Σ, μ') will have a measurement location. Let <math> f_1, f_2, \ ldots </ math> be a pointwise non-Reduced sequence of [0, ∞]-valued Σ- Measurable functions, eg ' all' k ≥ 1 and all x in X,
- <math> 0 \ leq f_k (x) \ leq f_ {k +1} (x). \, </ Mathematics>
Next, the pointwise limit of sequence <math> (f_ {n}) </ math> be f. That is, for every x in X,
- <math> F (x): = \ lim_ {k \ to \ infty} f_k (x). \, </ Mathematics>
Then f is Σ- Measurable and
- <math> \ Lim_ {k \ to \ infty} \ int f_k \, \ mathrm {d} \ mu = \ int f \, \ mathrm {d} \ mu. </ Mathematics>
'Remark. If the sequence <math> (f_k) </ math> satisfies the estimate' μ-almost everywhere, one can find some N ∈ Σ with μ ( N) = 0 such that the sequence <math> (f_k (x)) </ math> is non-Reduce all <math> x \ notin N </ math>. The advantage is that because all k,
- <math> \ Int f_k \, \ mathrm {d} \ mu = \ int_ {X \ backslash f_k N} \, \ mathrm {d} \ mu, \ \ text {and} \ \ int f \, \ mathrm {d} \ mu = \ int_ {X \ backslash N} f \, \ mathrm {d} \ mu </ math>
provided that f is Σ-Measurable (see example <ref name="SCHECHTER1997"> Erik Schechter (1997). Analysis and Its origins. </ ref> section 21.38).
Proof === === We first show that the f is Σ- Measurable (see Example [1] section 21.3). Do this, it is enough to show that the inverse image of an interval [0, t'] in f is the element of Sigma algebra Σ on X, because the (closed) intervals to the Borel Sigma algebra for reals. I' = [0, t'] be such a subinterval of [0, ∞]. Storage
- <math> F ^ {-1} (I) = \ {x \ in X \,). real test. theory of measure and integration.
{{cite book}}
: Check date values in:|year=
(help); Text "\, f (x) \ in I \}. </ Mathematics>
Since I' is a prolonged and close <math> \ forall k, f_k (x) \ Le f (x) </ math>,
- <math> F (x) \ in I \ Leftrightarrow f_k (x) \ in I, \ forall k \ in \ mathbb {N}. </ Mathematics>
Therefore,
- <math> \ {X \ in X \," ignored (help); line feed character in
|year=
at position 26 (help)CS1 maint: extra punctuation (link) CS1 maint: year (link) \ {x \ in X \, | \, f_k (x) \ in I \}. </ Math>
Note that each set in the countable intersection is a element of Σ because it is the inverse image of a Borel subset in a Σ- Measurable Function <math> f_k </ math>. Since Sigma algebras is, by definition, closed in countable intersection, this indicates that f is Σ-Measurable. In general, the supremum of the countable family of functions is also Measurable Measurable.
Now we will prove the rest of the monotone convergence Theorem. The fact that f is Σ-Measurable implies that information <math> \ int f \, \ mathrm {d} \ mu </ math> is well defined.
We begin by showing that <math> \ int f \, \ mathrm {d} \ mu \ geq \ lim_k \ int f_k \, \ mathrm {d} \ mu. </ Mathematics>
By means of the Lebesgue integral,
- <math> \ Int f \, \ mathrm {d} \ mu = \ sup \ {\ int g \, \ mathrm {d} \ mu \, | \, g \ in SF, \ g \ leq f \} </ math>
where SF is co-Measurable Σ simple functions on X. Since <math> f_k (x) \ leq f (x) </ math> of every x ∈ X, we have that
- <math> \ Left \ {\ int g \, \ mathrm {d} \ mu \, | \, g \ in SF, \ g \ leq f_k \ right \} \ subseteq \ left \ {\ int g \, \ mathrm {d} \ mu \, | \, g \ in SF, \ g \ leq f \ right \}. </ Math>
Hence, since the supremum of a subset can not be serious than all set, we have that:
- <math> \ Int f \, \ mathrm {d} \ mu \ geq \ lim_k \ int f_k \, \ mathrm {d} \ mu </ math>
and restrictions on political rights, since the sequence is monotonic.
We now prove the inequality in the other direction (which also follows from Fatou's Lemma), if we find that
- <math> \ Int f \, \ mathrm {d} \ mu \ leq \ lim_k \ int f_k \, \ mathrm {d} \ mu. </ Mathematics>
It follows from the definition of the integral, with a non-Decreasing sequence (g ''' k </ sub>) of non-not easy eight functions such that g k </ sub> ≤ f and such that
- <math> \ Lim_k \ int g_k \, \ mathrm {d} \ mu = \ int f \, \ mathrm {d} \ mu. </ Mathematics>
It suffices to prove that for all <math> k \ in \ mathbb {N} </ math>,
- <math> \ Int g_k \, \ mathrm {d} \ mu \ leq \ lim_j \ int f_j \, \ mathrm {d} \ mu </ math>
because if this is true for all k, then the limit of the left-hand side will be less than or equal to the right-hand side.
We will show that if g''' k </ sub> is a simple and Function
- <math> \ Lim_j f_j (x) \ geq g_k (x) \, </ math>
for every x, then
- <math> \ Lim_j \ int f_j \, \ mathrm {d} \ mu \ geq \ int g_k \, \ mathrm {d} \ mu. </ Mathematics>
Since the integral is linear, we may violate the Function <math> g_k </ math> in the constant value, reduce to the case in which <Math > g_k </ math> is the illustration of a Function element B of Sigma algebra Σ. In this case, we assume that <math> f_j </ math> is a sequence of functions which Measurable supremum of each point of B is more than or equal to a a.
To prove this result, fix ε> 0 and denote the sequence of Measurable hours
- <math> B_n = \ {x \ in B: f_n (x) \ geq 1 - \ epsilon \}. \, </ Mathematics>
By monotonicity of the integral, it says the following areas <math> n \ in \ mathbb {N} </ math>,
- <math> \ Mu (B_n) (1 - \ epsilon) = \ int (1 - \ epsilon) 1_ {B_n} \, \ mathrm {d} \ mu \ leq \ int f_n \, \ mathrm {d} \ mu </ math>
From that <math> \ lim_j f_j (x) \ geq g_k (x) </ math>, the x in B in <math> B_n </ math> enough for high value of n, and therefore
- <math> \ Bigcup_n B_n = B. </ Math>
Therefore, we have that
- <math> \ Int g_k \, \ mathrm {d} \ mu = \ int 1_B \, \ mathrm {d} \ mu = \ mu (B) = \ mu \ left (\ bigcup_n B_n \ right) . </ Math>
Using the monotonicity property of the measure, we can continue the above equalities as follows:
- <math> \ Mu \ left (\ bigcup_n B_n \ right) = \ lim_n \ mu (B_n) \ leq \ lim_n (1 - \ epsilon) ^ {-1} \ int f_n \, \ mathrm {d } \ mu. </ Mathematics>
Take' k → ∞, and the fact that this is true for any ε', the result follows.
See also == ==
Note == ==