Title: Theory / Unimportant / Math / Analysis / Real Pugh/{ / 6 - Lebesgue Theory
Tags: fromMath mathToDef automaticImplies complex hasSeenConjDef multipartite récupéré closed mathToSee CorrectType examplesSwitchExample1
On every note of this deck, unless a note says otherwise:
Chapter: 6

Name: The volume of \(I\)
Definition: \(\Pi_{i=1}^n(b_i-a_i)\).
Page: 384
Section: 4
Variables: \(I=\Pi_{i=1}^n(a_i,b_i)\)
Notation: \(|I|\)

Question: \(m^*\emptyset=\)
Answer: \(0\)
Page: 384
Chapter: 6
Section: 1
Index: 1a
Kind: Theo

Name: A Zero subset of \(\mathbb R^m\)
Definition: \(X\subseteq\mathbb R^m\) such that \(\mu^*(X)=0\).
Page: 386
Section: 1
Extra: 2 def
Typ: Lebesgue Measurable set

Name: The class of Zero sets
Page: 386
Section: 1
Chapter: 6
Index: 2
Kind: Prop
Under: taking subset
Under2: union
Condition2: countable

Question: \(\mu^*(X)=\)
Answer: \(0\).
Page: 386
Chapter: 6
Section: 1
Variables: \(X\) an hyperplane of \(\mathbb R^m\)
Index: 2
Kind: Prop

Name: A Lebesgue measurable subset of \(R^m\)
Definition: A subset \(E\subseteq R^m\) such that for all \(X\), \(m^*X =m^*(X\cap E)+m^*(X\setminus E^c)\).
Page: 388, 397
Section: 2, 4
Index: , 13
Kind: Theo
Example: An open or closed set
Name2: A measurable subset of \(R^m\)
Counterexample: A subset of \([0,1]\) which contains exactly one element of each equivalence class of the relation \(\{(x,y)\in \mathbb R^2:\frac xy\in\mathbb Q\}\)

Definition: The collection of all Lebesgue measurable subsets of \(\mathbb R^n\).
Page: 389
Section: 2
Notation: \(\mathcal M(\mathbb R^n)\)

Name: Measures on any set
Page: 389
Section: 2
Example: Cardinality
Example2: 0 on \(\emptyset\) and \(\infty\) everywhere else
Extra: Definition

Question: The collection of measurable sets is
Answer: a \(\sigma\)-algebra.
Page: 389
Chapter: 6
Section: 2
Variables: \(M\) a set and
\(\omega\) an outer measure on \(M\).
Index: 5
Kind: Theo

Smaller0: \(E\) is measurable
Greater0: equal to the measure of \(E\)
Chapter: 6
Section: 2
Variables: \(E,F\subseteq\mathbb R^m\)
Hide middle: yes
Prefix Greater: the measure of \(E\cup F\) is
Smaller1: \(F\) is a zero set

Name: A \(\sigma\)-algebra
Definition: contains the empty set
Section: 2
Definition2: closed under complement
Conjdef: A collection of sets that
Definition3: closed under countable union
French: Une tribu

Name: A meseomorphism between the \(M_i\)'s
Definition: which is a bijection
Page: 393
Section: 2
Variables: \((M_i,\omega_i)\) a measure set
\(\mathcal M_i\) the collection of measurable sets
Definition2: such that [$]\defFun{\mathcal M_1}{\mathcal M_2}{E}{f(E)}[/$] is a bijection
Conjdef: A function \(f:M_1\to M_2\)

Name: A meseometry between \(M_i\)'s
Definition: which is a meseometry
Page: 393
Section: 2
Variables: \((M_i,\omega_i)\) a measure set
\(\mathcal M_i\) the collection of measurable sets
Definition2: for each \(M\in\mathcal M_1\), \(\omega_1(M)=\omega_2(f(M))\)
Conjdef: A function \(f:M_1\to M_2\)

Smaller0: \(D\) is nonsingular diagonal
Greater0: to be a meseomorphism of Lebesgue mesure
Page: 394
Chapter: 6
Section: 2
Variables: \(D\) be an \(n\times n\) matrix.
Index: 8
Kind: Cor
Hide middle: yes
Prefix Greater: the linear map \(D:\mathbb R^n\to\mathbb R^n\) have the property

Smaller0: \(D\) is a nonsingular diagonal
Greater0: \(m(DE)=|\text{det} D|\,mE\)
Page: 394
Chapter: 6
Section: 2
Variables: \(D\) an \(n\times n\) matrix
Index: 8
Kind: Cor
Hide middle: yes
Smaller1: \(E\subseteq \mathbb R^n\) is measurable
Extra: diagonal?

Question: Mesure theoretical property of \(T\):
Answer: meseometry.
Page: 394
Chapter: 6
Section: 2
Variables: \(T\) a translation.

Question: In measure theory, \(T\) is
Answer: a meseomorphism.
Page: 394
Chapter: 6
Section: 2
Variables: \(T:\mathbb R^n\to\mathbb R^n\) an affine motion.
Index: 9
Kind: Theo

Smaller0: \(\mu^*(TS)\)
Greater0: \(|\text{det} T|\mu^*(S)\)
Page: 394
Chapter: 6
Section: 2
Variables: \(T:\mathbb R^n\to\mathbb R^n\) an affine motion
\(S\subseteq\mathbb R^n\).
Index: 9
Kind: Theo
Smaller to greater: =

Question: Mesure property of the boundary of a ball.
Answer: It is a zero-set.
Page: 394
Chapter: 6
Section: 2
Index: 10
Kind: Lem

Question: An open cube can be decomposed into
Answer: a countable disjoint union of open balls plus a zero set.
Page: 395
Chapter: 6
Section: 2
Index: 11
Kind: Lem

Name: A rigid motion
Definition: a translation
Page: 397
Section: 2
Variables: \(T\) an affine motion.
Index: 12
Kind: Corollary
Definition2: an orthogonal translation.
Conjdef: in this order
DefType: A meseometry

Definition: the class of countable union of closed sets
Page: 398
Section: 4
Notation: \(F_\sigma\)

Question: In \(F_\sigma\), \(\sigma\) means
Answer: «sum»
Page: 398
Chapter: 6
Section: 4

Smaller0: regular
Greater0: it can be sandwiched between an \(F_\sigma\) and a \(G_\delta\)-set, \(F\subseteq E\subseteq G\), such that \(m(G\setminus F)=0\)
Page: 399
Chapter: 6
Section: 4
Index: 16
Kind: Theorem
Smaller to greater: iff
Hide middle: yes
Smaller1: measurable
Prefix Smaller: Lebesgue measure is
Connect Smaller0: in the sense that a set \(E\) is

Name: The hull of \(A\)
Definition: A \(G_\delta\)-minimum set that contains \(A\)
Page: 400
Section: 4
Variables: \(n\in\mathbb N\)
\(A\subseteq\mathbb R^n\)
Abbreviation: \(H_A\)
Cardinal: Exactly one, up to zero-set

Name: The kernel of \(A\)
Definition: The greatest \(F_\sigma\)-set included in \(A\).
Page: 400
Section: 4
Variables: \(n\in\mathbb N\)
\(A\subseteq\mathbb R^n\).
Abbreviation: \(K_A\)
Cardinal: Exactly one, up to zero-set

Definition: The inner mesure of \(A\).
Page: 400
Section: 4
Variables: \(A\subseteq\mathbb R^n\).
Notation: \(m_*A\)

Smaller0: \(m_*\)
Greater0: \( m^*\)
Page: 400
Chapter: 6
Section: 4
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Smaller0: \(A\) is measurable
Greater0: \(m_*A=m^*A\)
Page: 400
Chapter: 6
Section: 4
Variables: \(A\subseteq\mathbb R^n\)
Smaller to greater: iff
Assuming: \(A\) is bounded
Extra: definition ?

Smaller0: \(m(A\times B)\)
Greater0: \(m(A)m(B)\)
Page: 401
Chapter: 6
Section: 5
Variables: \(A\subseteq \mathbb R^n, B\subseteq\mathbb R^k\).
Index: 21
Kind: Theo
Smaller to greater: =

Name: The undergraph of \(f\)
Definition: The set \(\{(x,y)\in\mathbb R\times [0,\infty): 0\le y<f(x)\}\).
Page: 406
Variables: \(f:\mathbb R\to[0,\infty)\).
Notation: \(\mathcal Uf\)

Name: A measurable function \(R\to[0,\infty)\).
Definition: A function \(R\to[0,\infty)\) such that \(\mathcal Uf\) is mesurable.
Page: 406
Kind: Definition

Question: \(\int f=\)
Answer: \(m(\mathcal U f)\)
Page: 406
Chapter: 6
Variables: \(f:\mathbb R\to[0,\infty)\) a mesurable function

Name: \(f_n\to f\) almost everywhere
Definition: \(f_n(x)\to f(x)\) except on a zero-set
Page: 407
Variables: \(f:\mathbb R\to[0,\infty)\).
Name2: \(f_n\to f\) presque partout
Abbreviation: \(f_n\to f\) a.e.
Abbreviation2: \(f_n\to f\) p.p.

Name: The completed undergraph of \(f\)
Definition: \[\{(x,y)\in\mathbb R\times [0,\infty):0\le y\le f(x)\}\].
Page: 407
Variables: \(f:\mathbb R\to[0,\infty)\).
Notation: \(\hat{\mathcal U}f\)

Smaller0: \({\hat{\mathcal U}}f\) is mesurable of measure \(x\)
Greater0: \({\mathcal U}f\) is mesurable of measure \(x\)
Page: 407
Chapter: 6
Variables: \(f:\mathbb R\to[0,\infty)\) and \(x\in[0,\infty)\).
Index: 28
Kind: Prop
Smaller to greater: iff
Hide middle: yes

Name: Lower (upper) envelope sequence of \((f_n)\)
Definition: \(\underline f_n(x)=\text{inf}\{f_k(x):k\ge n\}\) and \(\overline f_n(x)=\text{sup}\{f_k(x):k\ge n\}\) respectively.
Page: 408
Variables: \((f_n):\mathbb R\to[0,\infty)\).
Kind: Definition

Question: Example of unbounded convergent sequence of function such that limit and integral does not commute:
Answer: The sequence of steeple functions
Page: 409
Chapter: 6

Smaller0: \(f\le g\)
Greater0: \(\le\)
Page: 410
Chapter: 6
Variables: \(f,g:\mathbb R\to[0,\infty)\).
Index: 34
Kind: Theo
Prefix Greater: \(\int f\)
Suffix Greater: \(\int g\)

Smaller0: \(\int f\)
Greater0: \(\sum\int_{X_k} f\)
Page: 410
Chapter: 6
Variables: \(f:\mathbb R\to[0,\infty)\).
Index: 34 a
Kind: Theorem
Smaller to greater: \(=\)
Assuming: \(\mathbb R=\bigsqcup X_k\) and each \(X_k\) is measurable

Smaller0: \(\int\mathcal X_X\)
Greater0: \(m X\)
Page: 410
Chapter: 6
Variables: \(X\subseteq \mathbb R\)
Index: 34 c
Kind: Theorem
Smaller to greater: \(=\)
Assuming: \(X\) is measurable

Smaller0: \(X\) is measurable and \(mX=0\)
Greater0: \(=0\)
Page: 410
Chapter: 6
Variables: \(X\subseteq \mathbb R\),
\(f:\mathbb R\to[0,\infty)\).
Index: 34 d
Kind: Theorem
Prefix Greater: \(\int_X f\)

Question: \(\int cf\)
Answer: \(=c\int f\).
Page: 410
Chapter: 6
Variables: \(f:\mathbb R\to[0,\infty)\)
\(c\ge 0\).
Index: 34 f
Kind: Theorem
Assuming: \(f\) is measurable

Smaller0: \(\int(f+g)\)
Greater0: \(\int f+\int g\)
Page: 410, 413
Chapter: 6
Variables: \(f,g:\mathbb R\to[0,\infty)\).
Index: 34 h, 36
Kind: Theorem, cor
Smaller to greater: \(=\)
Assuming: \(f,g\) are measurable or integrable

Name: The \(f\)-translation
Definition: [$]\defFun{\mathbb R^2}{\mathbb R^2}{(x,y)}{(x,y+f(x))}[/$].
Page: 411
Variables: \(f:\mathbb R\to\mathbb R\)
Context: Analysis
Notation: \(T_f\)
Kind: Def

Question: \(T_f\circ T_g\) equals
Answer: \(T_{f+g}\)
Page: 411
Chapter: 6
Variables: \(f,g:\mathbb R\to\mathbb R\).

Definition: The functions which sends \(x\) to \(f(x)\) if \(f(x)\) is greater (respectively lesser) than \(0\) and \(0\) otherwise.
Page: 413
Variables: \((f_k):\mathbb R\to\mathbb R\).
Notation: \(f_+\) (respectively, \(f_-\))

Definition: \(\int f_+-\int f_-\).
Page: 414
Variables: \(f:\mathbb R\to\mathbb R\).
Notation: \(\int f\)

Question: Property of the set of \(f:\mathbb R\to\mathbb R\) measurable functions:
Answer: It is a vector space.
Page: 414
Chapter: 6
Index: 38
Kind: Prop

Question: Property of the set of \(f:\mathbb R\to\mathbb R\) integrable functions:
Answer: subspace of the space of measurable functions.subspace of the space of measurable functions.
Page: 414
Chapter: 6
Index: 38
Kind: Prop

Question: the function \(x\mapsto m(E_x)\) is
Answer: measurable
Page: 414
Chapter: 6
Section: 7
Variables: \(E\subseteq\mathbb R^k\times\mathbb R^n\).
Assuming: \(E\) is measurable
Name: Cavalieri's Principle
Extra: two principle with same name

Question: Almost every slice \(E_x\) of \(E\) is
Answer: measurable
Page: 414
Chapter: 6
Section: 7
Variables: \(E\subseteq\mathbb R^k\times\mathbb R^n\).
Assuming: \(E\) is measurable
Name: Cavalieri's Principle

Question: The integral of \(E\) is
Answer: [$$]mE=\int m(E_x)\dd x[/$$].
Page: 414
Chapter: 6
Section: 7
Variables: \(E\subseteq\mathbb R^k\times\mathbb R^n\).
Assuming: \(E\) is measurable
Name: Cavalieri's Principle

Definition: The \(x\)-slice \[\{y\in\mathbb R^n:(x,y)\in E\}\]
Page: 414
Section: 7
Variables: \(E\subseteq\mathbb R^k\times\mathbb R^n\),
\(x\in\mathbb R^k\)
Notation: \(E_x\)

Definition: The \(y\)-slice \[\{x\in\mathbb R^k:(x,y)\in E\}\]
Page: 414
Section: 7
Variables: \(E\subseteq\mathbb R^k\times\mathbb R^n\)
\(y\in\mathbb R^n \).
Notation: \(E^y\)

Question: As \(y\) increases, the \(y\)-slices of an undergraph
Answer: decrase monotonically
Page: 415
Chapter: 6
Section: 7
Index: 40
Kind: Cor

Definition: \(\bigcup _{y>a}(\mathcal Uf)^y\)
Page: 415
Section: 7
Variables: \(f:\mathbb R^k\times \mathbb R^n\to\mathbb R^k\),
\(a\in\mathbb R^k\)
Notation: \((\mathcal U f)^a\)
Index: 40
Kind: Cor

Definition: \(\bigcap _{y<a}(\hat{\mathcal U}f)^y\).
Page: 415
Section: 7
Variables: \(f:\mathbb R^k\times \mathbb R^n\to\mathbb R^k\),
\(a\in\mathbb R^k\)
Notation: \((\hat{\mathcal U} f)^a\)
Index: 40
Kind: Cor

Question: Every horizontal slice of a measurable undergraph is
Answer: measurable.
Page: 415,
Chapter: 6
Section: 7
Variables: \(f:\mathbb R^k\times \mathbb R^n\to\mathbb R^k\),
\(a\in\mathbb R^k\)
Index: 40
Kind: Cor

Smaller0: \(f\) is measurable
Greater0: [$]\int\int_{X\times Y} f(x,y)\dd x\dd y[/$]
Page: 416, 129
Chapter: 6 Pugh, 11 Zorich
Section: 7, 4.1
Variables: \(n,m\in\mathbb N\)
\(X\subseteq \mathbb R^m\), \(Y\subseteq \mathbb R^n\), intervals
\(f:\mathbb X\times Y\to[0,\infty)\)
Index: 42
Kind: Theo, Theo
Hide middle: yes
Name: Fubini-Tonelli Theorem
Greater1: [$]\int_{X}\dd x\int_{Y} f(x,y)\dd y[/$]
Connect Greater0: \(=\)

Smaller0: \(f\) is measurable
Greater0: a measurable function of \(y\)
Page: 416
Chapter: 6
Section: 7
Variables: \(f:\mathbb R^2\to[0,\infty)\)
Index: 42
Kind: Theo
Hide middle: yes
Prefix Greater: almost every slice \(y\mapsto f_x(y)\) is
Name: Fubini-Tonelli Theorem

Smaller0: \(f\) is measurable
Greater0: measurable
Page: 416
Chapter: 6
Section: 7
Variables: \(f:\mathbb R^2\to[0,\infty)\)
Index: 42
Kind: Theo
Hide middle: yes
Prefix Greater: the function [$]x\mapsto\int f_x(y)\dd y[/$] is

Smaller0: \(\mathcal V\) is composed of closed balls
Greater0: \(\mathcal V\) reduces to an efficient family of disjoint subcovering of almost all of \(A\)
Page: 418
Chapter: 6
Section: 8
Variables: \(A\subseteq\mathbb R^n\)
\(\mathcal V\) a Vitali covering of \(A\)
Index: 45
Kind: Lemma
Name: Vitali Covering Lemma

Name: The density of \(E\) at \(p\)
Definition: \(\lim_{Q\downarrow x}\frac{m(E\cap Q)}{mQ},\) if the limit exists.
Page: 422
Section: 8
Variables: \(E\subseteq \mathbb R^n\) be measurable
\(p\in\mathbb R^n\).
Notation: \(\delta(p,E)\)
Kind: Definition

Name: A density point of \(E\)
Definition: \(p\in\mathbb R^n\) such that \(\delta(p,E)=1\).
Page: 422
Section: 8
Variables: \(E\subseteq \mathbb R^n\) be measurable.

Name: The relative mesure or concentration of \(E\) in \(Q\)
Definition: \(\frac{m(E\cap Q)}{mQ}\).
Page: 422
Section: 8
Variables: \(E\subseteq \mathbb R^n\) be measurable.

Smaller0: \(E\) is measurable
Greater0: almost every \(p\in E\) is a density point of \(E\)
Page: 422
Chapter: 6
Section: 8
Variables: \(E\subseteq \mathbb R^n\) be measurable
Index: 47
Kind: Theorem
Hide middle: yes
Name: Lebesgue density theorem

Name: the set of density point of \(E\)
Page: 424
Section: 8
Variables: \(E\subseteq \mathbb R^n\) be measurable.
Intuition: the measure-theoric interior of \(E\).

Page: 424
Chapter: 6
Section: 8
Variables: \(E\subseteq \mathbb R^n\) be measurable.
Q: [...]
A: the set of density point of \(E^c\)
Q: [...]
A: measure-theoric exterior of \(E\)

Smaller0: \(f\) is locally integrable function and \(A\) have a finite positive measure
Greater0: average of \(f\)
Page: 426
Chapter: 6
Section: 9
Variables: \(f:\mathbb R^n\to\mathbb R\)
\(A\subseteq \mathbb R^n\) a measurable set.
Index: Definition
Prefix Greater: the
Greater1: [$$]\int_a f(x)\dd m=\frac{1}{mA}\int_Af(x)\dd m[/$$]
Connect Greater0: is

Smaller0: \(f\) is locally integrable
Greater0: for almost every \(p\in\mathbb R^n\) we have [$$]\lim_{Q\downarrow p}\int f(x)\dd m=f(p)[/$$]
Page: 426
Chapter: 6
Section: 9
Variables: \(f:\mathbb R^n\to\mathbb R\).
Index: 51
Kind: Theorem
Name: Average value theorem

Name: Locally integrable functions from \(\mathbb R^n\) to \(\mathbb R\)
Definition: A function which is integrable on a small enough neighborhood of each point.
Section: 9

Question: Property of \(F(x)\).
Answer: For almost every \(x\in [a,b]\) the derivative \(F'(x)\) exists and equals \(f(x)\).
Page: 428
Chapter: 6
Section: 9
Variables: \(f:[a,b]\to\mathbb R\) Lebesgue integrable
\(F(x)=\int_a^xf(t)dt\) its indefinite Lebesgue integral.
Index: 53
Kind: Cor

Name: Absolutely continuous real function
Definition: \[\sum_{i=1}^n b_i-a_i<\delta\implies \sum_{i=1}^n|G(b_i)-G(a_i)|<\epsilon.\]
Page: 429
Section: 9
Variables: \(G:[a,b]\to\mathbb R\).
Index: 54
Kind: Prop
Definition2: \[\sum_{i=1}^\infty b_i-a_i<\delta\implies \sum_{i=1}^\infty|G(b_i)-G(a_i)|<\epsilon.\]
Conjdef: For each \(\epsilon>0\) there exists \(\delta>0\) such that whenever \(([a_i,b_i])\) are disjoint intervals in \([a,b]\),
Definition3: \[\sum_{i=1}^\infty m([a_i,b_i])<\delta\implies \sum_{i=1}^\infty m([a_i,b_i])<\epsilon.\]
Definition4: \[\sum_{i=1}^n m([a_i,b_i])<\delta\implies \sum_{i=1}^n m([a_i,b_i])<\epsilon.\]

Smaller0: \(f\) is absolutely continuous
Greater0: \(fX\) is a zero set
Page: 429
Chapter: 6
Section: 9
Variables: \(f:[a,b]\to\mathbb R\)
\(X\subseteq[a,b]\).
Index: 54
Kind: Prop
Hide middle: yes
Smaller1: \(X\) is a zero set
Connect Smaller0: and

Smaller0: \(f\) is absolutely continuous
Greater0: there exists \(\delta>0\) such that if \(E\) is measurable then \(fE\) is measurable and \(mE<\delta\implies m(fE)<\epsilon\)
Page: 429
Chapter: 6
Section: 9
Variables: \(f:[a,b]\to\mathbb R\)
\(\epsilon>0\)
Index: 54
Kind: Prop
Hide middle: yes

For almost every \(x\) the derivative \(F'(x)\) exists and equals \(f(x)\).
\(F\) is absolutely continuous.
If \(G\) is an absolutely continuous function and \(G'(x)=f(x)\) for almost every \(x\) then \(G\) differs from \(F\) by a constant.
Page: 430
Chapter: 6
Section: 9
Variables: \(f:[a,b]\to\mathbb R\) be Lebesgue integrable
\(F\) be its indefinite integral \(F(y)=\int_a^x f(t)dt\).
Index: 55
Kind: Theorem
Name: Lebesgue's main theorem

Smaller0: \(f\) is monotone
Greater0: \(f\) has a derivative almost everywhere
Page: 433
Chapter: 6
Section: 10
Variables: \(f:[a,b]\to\mathbb R\).
Index: 57, 59
Kind: Theorem
Hide middle: yes
Smaller1: \(f\) is Lipschitz
Extra: 538
Connect Smaller0: or

Name: variation of \(f\)
Definition: \(\sum_{k=1}^n|\Delta_k f|\).
Page: 438
Section: 10
Variables: \(f:[a,b]\to\mathbb R\) and \(X:a=x_0,\dots,x_n=b\) a partition of \([a,b]\).

Smaller0: \(f\) has bounded variation
Greater0: \(f\) is a.e. differentiable
Page: 438
Chapter: 6
Section: 10
Variables: \(f:[a,b]\to\mathbb R\).
Index: 60
Kind: Theo
Hide middle: yes

Smaller0: \(f\) is absolutely continuous
Greater0: \(f\) has bounded variation
Page: 438
Chapter: 6
Section: 10
Variables: \(f:[a,b]\to\mathbb R\).
Index: 61, 62
Kind: Theo, Cor
Hide middle: yes
Greater1: \(f\) is almost everywhere differentiable
Connect Greater0: and
Extra: , 439

Smaller0: \(E\) is measurable and has positive mesure
Greater0: there exists \( \delta>0\) such that for all \(t\in(-\delta,\delta)\), the \(t\)-translate of \(E\) meets \(E\)
Page: 441
Chapter: 6
Section: B
Variables: \(E\subseteq \mathbb R\).
Index: 64
Kind: Theorem
Hide middle: yes
Name: Steinhaus' Theorem

Question: The circle splits into
Answer: two nonmeasurable disjoint subsets that each has inner measure zero and outer measure one.
Page: 442
Chapter: 6
Section: B
Index: 66
Kind: Theorem

Smaller0: \(E\) has a positive measure
Greater0: there exists \(N\subseteq E\) such that for all \(E'\subseteq E\) measurable, \(m^*(N\cap E')=mE'\) and \(m_*N=0\).
Page: 442
Chapter: 6
Section: B
Variables: \(E\subseteq \mathbb R^n\).
Index: 67
Kind: Cor
Hide middle: yes

Name: An affine motion
Definition: An invertible linear transformation, followed by a translation.
Page: 394
Section: 3
Variables: \(n\in\mathbb N\)
\(T:\mathbb N^n\to\mathbb N^n\).

Q: The [...] is [...]
A: total variation of \(f\); the supremum of the variations over all partitions
\(X\).
Page: 438
Chapter: 6
Section: 10
Variables: \(f:[a,b]\to\mathbb R\).

Name: A function of bounded variation
Definition: A function whose total variation of \(f\) is finite
Page: 438
Section: 10
Variables: \(f:[a,b]\to\mathbb R\).

Name: The Lebesgue outer measure of \(A\)
Definition: \(\inf\left\{ \sum_{k\in\mathbb N} |I_k|: \{I_k\} \text{ is a covering of $A$ by open boxes of $\mathbb R^n$}\right\}\).
Page: 383
Section: 4
Variables: \(A\subseteq\mathbb R^n\).
Notation: \(m^*A\)

Name: A monotone function on \(\mathbb R^m\)
Definition: A function \(m\) such that \(A\subseteq B\) implies \(mA\le mB\).
Page: 384
Section: 1
Variables: \(A\subseteq B\subseteq \mathbb R^m\).
Kind: Theo

Smaller0: \(E\) is a zero set
Greater0: almost every slice of \(E\) is a (slice) zero set
Page: 403
Chapter: 6
Section: 5
Variables: \(E\subseteq \mathbb R^n\times \mathbb R^k\).
Index: 26
Kind: Theorem
Smaller to greater: iff
Assuming: \(E\) is measurable

Smaller0: \(g\) is the pointwise limit of \((f_n)\)
Greater0: \(g\) is measurable
Page: 410
Chapter: 6
Variables: \((f_n),g:X\to[0,\infty)\).
Index: 32
Kind: Cor
Hide middle: yes

Smaller0: increases outer measure by at most a factor \(t\) and its inverse increases outer measure by at most a factior \(1/t\)
Greater0: meseomorphism
Page: 393
Chapter: 6
Section: 2
Index: 7
Kind: Theo
Prefix Greater: it is a
Prefix Smaller: a bijection

Smaller0: \(X\in G_\delta\)
Greater0: \(X\) belongs to \(f(X)\in G_\delta\)
Page: 399
Chapter: 6
Section: 4
Variables: \(f\) an homeomorphism and \(X\subseteq\mathbb R^n\).
Hide middle: yes

Smaller0: the \(f_n\) are measurable
Greater0: \(\int f_n\uparrow\int f\)
Page: 407
Chapter: 6
Variables: \(f,(f_n):\mathbb R\to[0,\infty)\).
Index: 27
Kind: Theo
Hide middle: yes
Smaller1: \(f_n\uparrow f\) a.e. as \(n\to\infty\)
Name: Monotone Convergence Theorem

Smaller0: \(f_n\) are measurable
Greater0: \(\int f_n\downarrow\int g\)
Page: 408
Chapter: 6
Variables: \((f_n),g:\mathbb R\to[0,\infty)\).
Index: 29
Kind: Cor
Hide middle: yes
Smaller1: \(f_n\downarrow g\) a.e. as \(n\to\infty\)

Smaller0: the \(f_n\) are measurable,
Greater0: \(h\) is integrable and \(\int\lim f_n=\lim\int f\)
Page: 409
Chapter: 6
Variables: \((f_n),g,h:\mathbb R\to[0,\infty)\).
Index: 31
Kind: Theo
Smaller1: \(\lim f_n\) exists a.e.,
Name: Dominated convergence theorem
Smaller2: \(\int g<\infty\)
Smaller3: \(f_n\)

Smaller0: the \(f_n\) are measurable functions
Greater0: \(\int\text{lim inf}f_n\le\text{lim inf}\int f_n\)
Page: 410
Chapter: 6
Variables: \((f_n):\mathbb R\to [0,\infty)\).
Index: 33
Kind: Theo
Name: Fatou's Lemma

Smaller0: \(\mu^*(\bigcup A_i)\)
Greater0: \( \sum\mu^* A_i\)
Page: 384
Chapter: 6
Section: 1
Variables: \((A_n)\subseteq\mathbb R^m\).
Smaller to greater: \(\le\)
Name: A subadditive function on \(\mathbb R^m\)
Greater to smaller: \(\ge\)

Name: An abstract outer measure
Definition: its image on the emptyset is 0,
Page: 389
Section: 2
Variables: \(M\) a set.
Definition2: countably subadditive
Conjdef: equivalenty
Definition3: monotone.
Denoted by: \(\omega\)
French: Mesure extérieure
DefType: \(2^M\to[0,\infty]\) such that

Definition: The class of countable intersection of open sets.
Page: 398
Section: 4
Notation: \(G_\delta\)

Question: In \(G_\delta\), the \(\delta\) means
Answer: durschnitt.
Page: 398
Chapter: 6
Section: 4

Question: Every indefinite integral is
Answer: absolutely continuous.
Page: 431
Chapter: 6
Section: 9
Index: 56
Kind: Theorem

Question: Every absolutely continuous function has
Answer: a derivative almost everywhere and up to a constant it is the indefinite integral of its derivative.
Page: 431
Chapter: 6
Section: 9
Index: 56
Kind: Theorem
Name: Lebesgues's Antiderivative Theorem

Name: An inner regular measure
Definition: A measure such that \(\mu(A)=\text{sup}\{\mu(F)\mid F\subseteq A, F\text{ compact and measurable}\}\) for each \(A\) measurable
Page: 399
Section: 4
Variables: \(X\) a set
\(T\) a topology over \(X\)
\(\Sigma\) a \(\sigma\)-algebra on \(X\)
Index: 16
Kind: Theo

Name: An outer regular measure
Definition: A measure such that \(\mu(A)=\text{inf}\{\mu(F)\mid G\supseteq A, F\text{ open and measurable}\}\) for each \(A\) measurable
Page: 399
Section: 4
Variables: \(X\) a set
\(T\) a topology over \(X\)
\(\Sigma\) a \(\sigma\)-algebra on \(X\)
Index: 16
Kind: Theo
French: Mesure extérieure

Name: A regular measure
Definition: inner and outer regular
Page: 399, wp
Section: 4
Variables: \(X\) a set
\(T\) a topology over \(X\)
\(\Sigma\) a \(\sigma\)-algebra on \(X\)
Index: 16
Kind: Theo
Definition2: for each \(E\) measurable, there exists \(F\) a \(F_\sigma\)-set, \(G\) a \(G_\delta\)-set and \(F\subseteq E\subseteq G\) such that \(m(G\setminus F)=0\)
Conjdef: A measure \(m\) which is, equivalently

Name: A Vitali covering
Definition: for each point \(p\in A\),
Page: 418
Section: 8
Variables: \(M\) a metric space
\(A\subseteq M\)
Kind: Def
Definition2: for each \(r>0\)
Conjdef: A covering such that
Definition3: there is \(V\in\mathcal V\) such that
Definition4: \(\{p\}\subsetneq V\subseteq M_rp\) and
Denoted by: \(\mathcal V\)

Name: A disjoint covering of almost all of \(A\)
Definition: the elements of \(\mathcal V\) are disjoint
Page: 418
Section: 8
Variables: \(M\) a metric space
\(A\subseteq M\)
Kind: Def
Definition2: \(A\setminus\bigsqcup_{V\in\mathcal V}V\) is a zero set
Conjdef: A set \(\mathcal V\) of subsets of \(A\) such that

Name: An efficient family of disjoint covering of almost all of \(A\)
Definition: A class \(\mathfrak V\) of disjoint covering of alsmost all of \(A\), such that, for any \(\epsilon>0\), \(\mathfrak V\) contains a class \(\mathcal V\) such that \(m\bigsqcup _{V\in\mathcal V}V\le m^*A+\epsilon\)
Page: 418
Section: 8
Variables: \(M\) a metric space
\(A\subseteq M\)
Kind: Def

Question: Why is the Aleph positionned where it is, in an attique of a house soon to be destroyed ?
Answer: Because such an extraordinary things must be in the most mundane place, in order not to introduce multiple wonderful things

Question: complement of the set of density point of \(E\) and of \(E^c\) is
Answer: the measure-theoric boundary of \(E\)
Page: 424
Chapter: 6
Section: 8
Variables: \(E\subseteq \mathbb R^n\) be measurable.