Title: Theory / Unimportant / Math / Measure theory - Fremlin{ / 1 Introduction
Tags: fromMath mathToDef fromImplication récupéré hasSeenConjDef closed mathToSee TypToProperty examplesSwitchExample1 multipartite property TypToQuestion counterExample fromRelation mathToDo mid1516974248815 omitted toUnary

Name: A \(\sigma\)-algebra
Definition: containing the empty set,
Page: 10
Chapter: 1
Section: 11A
Variables: \(X\) a set
Context: Family of subsets of \(X\)
Kind: Def
Definition2: closed under countable union.
Example: \(\{\emptyset,X\}\),
Conjdef: A family of subsets of \(X\)
Definition3: closed under complement
Name2: A \(\sigma\)-field
Example2: \(\mathcal PX\), the set of subsets of \(X\)
French: Une tribu

Name: A measurable space
Definition: Some pair \((X,\Sigma)\) with \(X\) a set and \(\Sigma\) a \(\sigma\)-algebra of subsets of \(X\).
Page: 10
Chapter: 1
Section: 11B
Index: (c)

Name: Symmetric difference
Definition: Sending pairs \((E,F)\) of sets to the set of elements either in \(E\) or in \(F\) but not in both.
Page: 10
Chapter: 1
Section: 11C
Context: set
Notation: \(\Delta\)

Name: Set difference
Definition: Sending pairs \((E,F)\) of sets to the set of elements belonging to \(E\) and not to \(F\).
Page: 10
Chapter: 1
Section: 11C
Context: set theory
Notation: \(\backslash\)

Name: \(\Sigma\)
Page: 10, 11
Section: 11C
Chapter: 1
Index: b c d
Variables: \(X\) a set
\(\Sigma\) a \(\sigma\)-algebra
Under: union
Condition: countable
Under2: set difference
Under3: intersection
Under4: complement

Name: Countable set
Definition: A set \(X\) such that there is a surjective function from \(\mathbb N\) to \(X\).
Page: 12
Chapter: 1
Section: 11F
Example: \(\mathbb Z\)
Example2: \(\mathbb Q\)
Example3: \(\mathbb N\)

Name: Countable sets
Page: 12
Section: 11F
Chapter: 1
Index: b i ii
Under: subset
Under2: finite cartesian product
Under3: countable union

Name: Bijection from \(\mathbb N\) to \(\mathbb Z\)
Page: 11
Chapter: 1
Section: 11E
Example: \(n\mapsto(-1)^n\left\lfloor\frac{n+1}{2}\right\rfloor\)

Question: Bijection from \(\mathbb N\) to \(\mathbb Q\)
Answer: \(n\mapsto\frac{n-m^3-m^2}{m+1}\) with \(m=\left\lfloor\sqrt[3]{n}\right\rfloor\)
Page: 11
Chapter: 1
Section: 11E
Extra: Firt values are:
[(0, 1), (-1, 2), (0, 2), (1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (-4, 3), (-3, 3), (-2, 3), (-1, 3), (0, 3), (1, 3), (2, 3), (3, 3), (4, 3), (5, 3), (6, 3), (7, 3)], so not a bijection

Name: The set of \(\sigma\)-algebra of subsets of \(X\)
Page: 12
Section: 11G
Chapter: 1
Index: (a)
Variables: \(X\) a set
Under: arbitrary intersection

Name: The Borel sets of \(\mathbb R^n\)
Definition: \(\Sigma_{\text{set of open sets of $\mathbb R^n$} }\)
Page: 13, 41
Chapter: 11, 12
Section: 1G, 1J
Variables: \(n\in\mathbb N\)
Index: (c)
Definition2: The \(\sigma\)-algebra generated by subsets of \(\mathbb R^n\) whose \(i\)-th component is less than or equal to \(\alpha\), for \(i\) and \(\alpha\) fixed

Name: A measure space
Definition: \((X,\Sigma)\) a measurable space
Page: 14
Chapter: 1
Section: 12A
Definition2: \(\mu\) a measure on \(X\).
Conjdef: A triple \((X,\Sigma, \mu)\) with

Name: A measurable set
Definition: An element of \(\Sigma\).
Page: 14
Chapter: 1
Section: 12A
Variables: \((X,\Sigma, \mu)\) a measure space

Name: A measure on \(X\)
Definition: \(\mu(\emptyset)=0\)
Page: 14
Chapter: 1
Section: 12A
Variables: \((X,\Sigma)\) a measurable space
Definition2: for each disjoint sequence \((E_n)_{n\in\mathbb N}\), \(\mu(\bigcup_n E_n)=\sum_{n}\mu(E_n)\).
Conjdef: both
DefType: A function \(\mu:\Sigma\to [0,\infty]\),

Question: \(\infty+\infty\) is
Answer: \(\infty\)
Page: 15
Chapter: 1
Section: 12B

Question: \(\infty-\infty\) is
Answer: not defined
Page: 15
Chapter: 1
Section: 12B

Question: \(\infty-a\) is
Answer: \(\infty\)
Page: 15
Chapter: 1
Section: 12B
Variables: \(a\in\mathbb R\)

Question: \(\infty+a\) is
Answer: \(\infty\)
Page: 15
Chapter: 1
Section: 12B
Variables: \(a\in\mathbb R\)

Question: \(\infty\cdot a\) is
Answer: \(\infty\).
Page: 15
Chapter: 1
Section: 12B
Variables: \(a\in\mathbb R\)
Assuming: \(a\ge 0\)

Question: \(\text{inf}\emptyset\) is
Answer: \(\infty\).
Page: 15
Chapter: 1
Section: 12B

Name: A disjoint set of subsets of \(X\)
Definition: A set \(\mathcal S\subseteq\mathcal PX\) such that, for each \(X,Y\in \mathcal S\), \(X\) and \(Y\) are disjoint.
Page: 15
Chapter: 1
Section: 12B
Variables: \(X\) a set,

Name: Two disjoint sets
Definition: \(A,B\) sets such that \(A\cap B=\emptyset\).
Page: 15
Chapter: 1
Section: 12B
Variables: \(X\) a set,

Name: A point-supported measure
Definition: A measure of the form \(E\mapsto \text{sup}\{\sum_{x\in I}h(x)\mid I\subseteq E\}\) where \(h:X\to [0,\infty]\) is an arbitrary function.
Page: 15
Chapter: 1
Section: 12B(d)
Variables: \(X\) a set

Name: The counting measure
Definition: A measure on \(\mathcal PX\) associating to each set its cardinality.
Page: 15
Chapter: 1
Section: 12B(d)
Variables: \(X\) a set,

Name: The Dirac measure on \(X\) concentrated at \(x\)
Definition: The point-supported measure sending \(x\) to 1 and every other element of \(X\) to 0.
Page: 15
Chapter: 1
Section: 12B(d)
Variables: \(X\) a set,
\(x\in X\)

Name: Finite measure on \(\mathbb N\)
Page: 15
Chapter: 1
Section: 12B(d)
Example: The point supported measure \(n\mapsto 2^{-n-1}\)

Smaller0: \(\mu(E\cup F)\)
Greater0: \(\mu E+\mu F\)
Page: 15
Chapter: 1
Section: 12C
Variables: \((X,\Sigma,\mu)\) a measure space,
\(E,F\in\Sigma\)
Index: (a)
Smaller to greater: \(=\)
Assuming: \(E\) is disjoint from \(F\)

Smaller0: \(E\subseteq F\)
Greater0: \(\le\)
Page: 15
Chapter: 1
Section: 12C
Variables: \((X,\Sigma,\mu)\) a measure space,
\(E,F\in\Sigma\)
Prefix Greater: \(\mu E\)
Suffix Greater: \(\mu F\)

Smaller0: \(\mu(E\cup F)\)
Greater0: \(\mu E+\mu F\)
Page: 15
Chapter: 1
Section: 12C
Variables: \((X,\Sigma,\mu)\) a measure space,
\(E,F\in\Sigma\)
Index: (c)
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Smaller0: \(\mu(\bigcup _nE_n)\)
Greater0: \(\sum_n \mu E_n\)
Page: 16
Chapter: 1
Section: 12C
Variables: \((X,\Sigma,\mu)\) a measure space,
\((E_n)_{x\in \mathbb N}\in\Sigma\)
Index: (d)
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Smaller0: \((E_n)\) non decreasing
Greater0: \(\mu(\bigcup _nE_n)\)
Page: 16
Chapter: 1
Section: 12C
Variables: \((X,\Sigma,\mu)\) a measure space,
\((E_n)_{x\in \mathbb N}\in\Sigma\)
Index: (e)
Greater1: \(\lim _{n}\mu E_n\)
Connect Greater0: \(= \)

Smaller0: \((E_n)\) non increasing
Greater0: \(\mu(\bigcap _nE_n)\)
Page: 16
Chapter: 1
Section: 12C
Variables: \((X,\Sigma,\mu)\) a measure space,
\((E_n)_{x\in \mathbb N}\in\Sigma\)
Index: (f)
Greater1: \(\lim _{n}\mu E_n\)
Connect Greater0: \(= \)

Name: A (\(\mu\))-negligible set
Definition: there exists \(E\in\Sigma\) such that \(A\subseteq E\) and \(\mu E=0\).
Page: 16, 59
Chapter: 1
Section: 12D, 32A
Variables: \((X,\Sigma,\mu)\) a measure space,
Index: (a), (d)
Definition2: \(\mu^*A=0\)
Conjdef: A set \(A\subseteq X\) such that
Name2: A null set

Name: The null ideal of the measure \(\mu\)
Page: 16
Section: 12D
Chapter: 1
Index: (b)
Variables: \((X,\Sigma,\mu)\) a measure space,
Under: subset
Closure2: contain
Under2: \(\emptyset\)
Under3: union
Condition3: finite
Condition4: countable

Name: The null ideal of the measure \(\mu\)
Definition: The set of negligible set.
Page: 16
Chapter: 1
Section: 12D
Variables: \((X,\Sigma,\mu)\) a measure space,
Index: (b)

Name: A \(\sigma\)-ideal of sets
Definition: A set of subset of \(X\), containing the empty set, closed under subset and countable union.
Page: 16
Chapter: 1
Section: 12D
Variables: \(X\) a set
Index: (b)

Name: A conegligible set
Definition: A set whose complement is negligible.
Page: 16
Chapter: 1
Section: 12D
Variables: \((X,\Sigma,\mu)\) a measure space,
Index: (b)

Name: \(P(x)\) for (\(\mu\))-almost every \(x\in X\)
Definition: \(P\) is conegligible.
Page: 16
Chapter: 1
Section: 12D
Variables: \((X,\Sigma,\mu)\) a measure space,
\(P\) a property defined on \(X\)
Index: (b)
Name2: \(P\) (\(\mu\))-almost surely
Abbreviation: \(P(x)\) (\(\mu\))-a.e. (x)
Name3: \(P\) (\(\mu\))-presque partout
Abbreviation2: \(P\) (\(\mu\))-a.s.
Name4: \(P\) (\(\mu\))-almost everywhere
Abbreviation3: \(P\) (\(\mu\))-p.p.

Name: An outer measure on \(X\)
Definition: subadditive
Page: 19
Chapter: 1
Section: 13A
Variables: \(X\) a set
Kind: Def
Definition2: monotonic
Conjdef: function \(\theta:\mathcal PX\to[0,\infty]\)
Definition3: \(\theta\emptyset =0\)
French: Mesure extérieure sur \(X\)

Name: A complete measure space
Definition: A measure space whose negligible sets are measurable.
Page: 17
Chapter: 1
Section: 12F
Kind: Def
Intuition: When a set have measure 0, its subsets also have measure 0

Name: Carathéodory’s Method
Definition: The set of sets \(E\subseteq X\) such that \(\theta A=\theta (A\cap E)+\theta(A\setminus E)\) for every \(A\subseteq X\).
Page: 19
Chapter: 1
Section: 13C
Variables: \(X\) a set,
\(\theta\) an outer measure on \(X\)
Kind: Def
Typ: A \(\sigma\)-algebra of subsets of \(X\)

Question: Example of measure on the \(\sigma\)-algebra of subsets of \(X\) generated by the Carathéodory’s Method
Answer: \(\theta\)
Page: 19
Chapter: 1
Section: 13C
Variables: \(X\) a set,
\(\theta\) an outer measure on \(X\)
Kind: Def

Name: A half-open interval in \(\mathbb R^r\) with bounds \(\vec a,\vec b\)
Definition: A set of the form \(\{(x_1,\dots,x_r)\mid a_i\le x_i< b_i\}\).
Page: 29
Chapter: 1
Section: 15B
Variables: \(r\in\mathbb N^+\)
\(a_1,b_1,\dots, a_r,b_r\in\mathbb R\)
Notation: \([\vec a,\vec b[\)
Kind: Def
Extra: Section 14 omitted. It is a one dimensional version of section 15

Name: The volume of \([\vec a,\vec b[\)
Definition: \(\Pi (b_i-a_i)\).
Page: 29
Chapter: 1
Section: 15B
Variables: \(r\in\mathbb N^+\)
\(a_1,b_1,\dots, a_r,b_r\in\mathbb R\)
Notation: \(\lambda([\vec a,\vec b[)\)
Kind: Def

Name: The Lebesgue outer measure on \(\mathbb R^r\)
Definition: The function sending \(A\subseteq\mathbb R^r\) to the infimum of \(\sum_j\lambda I_j\), where \((I_j)\) is a countable covers \(A\) and \(\lambda\) is the volume function.
Page: 30, 60
Chapter: 11, 13
Section: 5, 2
Variables: \(r\in\mathbb N^+\)
Index: , C
Kind: Def
Definition2: \(\mu^*\) where \(\mu\) is the Lebesgue measure

Name: The Lebesgue measure on \(\mathbb R^r\)
Definition: The measure obtained by Carathéodory's method using the Lebesgue outer measure on \(\mathbb R^r\).
Page: 31
Chapter: 1
Section: 15E
Variables: \(r\in\mathbb N^+\)
Kind: Def

Name: A Lebesgue measurable set
Definition: A set belonging to the \(\sigma\)-algebra obtained by Carathéodory's method using the Lebesgue outer measure on \(\mathbb R^r\).
Page: 31
Chapter: 1
Section: 15E, F, G
Variables: \(r\in\mathbb N^+\)
Kind: Def

Name: The subspace \(\sigma\)-algebra of subsets of \(D\)
Definition: \(\{E\cap D\mid E\in\Sigma\}\).
Page: 35
Chapter: 12
Section: 1A
Variables: \(X\) a set,
\(\Sigma\) a \(\sigma\)-algebra of subsets of \(X\),
\(D\subseteq X\)
Notation: \(\Sigma_D\)
Kind: Lem
Name2: The trace of \(\Sigma\) on \(D\)
DefType: \(\sigma\)-algebra of subsets of \(D\)

Name: A (\(\Sigma\)-)measurable real-valued function
Definition: \(\{x\mid f(x)\sim a\}\in\Sigma_D\) for all \(a\in\mathbb R\), with \(\sim\) being any or all of the following symbols: \(<\), \(\ge\), \(\le\) or \(>\).
Page: 36
Chapter: 12
Section: 1C
Variables: \(X\) a set,
\(\Sigma\) a \(\sigma\)-algebra of subsets of \(X\),
Kind: Def
Conjdef: A function \(f:D\to\mathbb R\), for some \(D\subseteq X\), which satisfies the following property

Smaller0: \(g\) is continuous
Greater0: \(g\) is Borel measurable
Page: 36
Chapter: 12
Section: 1D
Variables: \(r\in\mathbb N^+\),
\(D\subseteq\mathbb R^r\),
\(g:D\to\mathbb R\)
Index: (a)
Kind: prop
Smaller1: \(g\) is monotonic (r=1)
Extra: rieman ?
Connect Smaller0: or
Greatest: \(g\) is Lebesgue measurable

Name: The set of Lebesgue measurable real/complex-valued function
Page: 36, 39, 41, 43, 64
Section: 1E, H, 1J, notes, E
Chapter: 12, 13
Kind: Theo, Prop, Lem
Extra: Counter example for composition: later
Under: sum, product, and quotient when defined
Under2: limit of functions, where it is defined
Under3: lim inf and lim sup
Condition3: sequence of functions
Under4: minimum and maximum of a set of functions
Condition4: countable
Under5: composition
Condition5: on the left by applying a \(n\)-ary Borel measurable function to \(n\) measurable functions
Under6: restricting the domain of definition
Under7: multiplication
Condition7: a real/complex constant
Under8: contains constant functions

Question: \(f^{-1}[E]\) is
Answer: the intersection of \(\text{dom } f\) and of a \(\Sigma\)-measurable set.
Page: 37
Chapter: 12
Section: 1E
Variables: \((X,\Sigma)\) a measurable space,
\(f:X\rightharpoondown\mathbb R\) measurable,
\(E\subseteq \mathbb R\) a Borel set
Index: (f)
Kind: Theo

Definition: \(x\mapsto\lim_{n\to\infty}f_n(x)\) defined on the subset of \(\bigcup_{n\in\mathbb N}\bigcap_{m\ge n}\text{dom }f_n\) where the limit exists.
Page: 38
Chapter: 12
Section: 1F
Variables: \((X,\Sigma)\) a measurable space,
\((f_n)_{n\in\mathbb N}:X\rightharpoondown\mathbb R\) \(\Sigma\)-measurable,
Notation: \((\lim_{n\to\infty}f_n)\)
Index: (a)
Kind: Theo
DefType: \(\Sigma\)-measurable

Definition: \(x\mapsto \text{sup}_{n\to\infty}f_n(x)\) defined on the subset of \(\bigcap_{n\in\mathbb N}\text{dom }f_n\) where the limit exists.
Page: 38
Chapter: 12
Section: 1F
Variables: \((X,\Sigma)\) a measurable space,
\((f_n)_{n\in\mathbb N}:X\rightharpoondown\mathbb R\) measurable,
Notation: \((\text{sup} _{n\to\infty}f_n)\)
Index: (b) (c)
Kind: Theo
DefType: \(\Sigma\)-measurable

Definition: \(x\mapsto\lim \text{sup}_{n\to\infty}f_n(x)\) defined on the subset of \(\bigcup_{n\in\mathbb N}\bigcap_{m\ge n}\text{dom }f_n\) where the limit exists.
Page: 39
Chapter: 12
Section: 1F
Variables: \((X,\Sigma)\) a measurable space,
\((f_n)_{n\in\mathbb N}:X\rightharpoondown\mathbb R\) measurable,
Notation: \((\lim\text{sup} _{n\to\infty}f_n)\)
Index: (d) (e)
Kind: Theo
DefType: \(\Sigma\)-measurable

Smaller0: \(f\) is measurable
Greater0: measurable and extending \(f\)
Page: 40
Chapter: 12
Section: 1I
Variables: \((X,\Sigma)\) a measurable space,
\(D\subseteq X\),
\(f:D\to\mathbb R\)
Kind: Prop
Smaller to greater: iff
Prefix Greater: there is a function \(h:X\to \mathbb R\)
Intuition: keeps measurability.
When creating a superset, we can essentially ignore everything else

Name: The characteristic function of \(A\)
Definition: [$]\defFun{X}{\{0,1\}}{x}{\left\{\begin{array}{ll}1&\text{if $x\in A$}\\0&\text{otherwise}\end{array}\right.}[/$].
Page: 43
Chapter: 12
Section: 2A
Variables: \(X\) a set,
\(A\subseteq X\)
Notation: \(\Xi^A\)
Index: (a)
Kind: Def

Smaller0: \(A\in\Sigma\)
Greater0: measurable
Page: 43
Chapter: 12
Section: 2A
Variables: \((X,\Sigma)\) a measurable set,
\(A\subseteq X\)
Index: (a)
Kind: Def
Smaller to greater: iff
Prefix Greater: \(\Xi^A\) is

Name: A simple function
Definition: of the form \(\sum_{i=0}^n a_i\Xi^{E_i}\), with \(a_i\in\mathbb R\) and \(E_i\in \Sigma\) of finite measure.
Page: 44
Chapter: 12
Section: 2A, B, C
Variables: \((X,\Sigma,\mu)\) a measure,
Index: (b), (a), (b)
Kind: Def, Lem, Lem
Definition2: \(E_i\)'s are disjoint
Conjdef: A function
Typ: Measurable

Name: The set of simple function
Page: 44
Chapter: 12
Section: 2B
Variables: \((X,\Sigma)\) a measurable set,
Index: (b)(c)(d)
Kind: Lem
Typ: A real vector space

Smaller0: the \(E_i\)'s are measurable
Greater0: a finite union of some disjoint measurable sets \(\bigcap _{E\in P}E\setminus\bigcup_{E\in \mathcal E\setminus P}\bigcup E\) for \(P\subseteq\mathcal E\),
Page: 44
Chapter: 12
Section: 2C
Variables: \((X,\Sigma,\mu)\) a measure set,
\(\mathcal E=\{E_0,\dots,E_n\}\)
Index: (a)
Kind: Lem
Prefix Greater: the \(E_i\)'s can be written canonically as

Smaller0: \(\sum_{i=0}^na_i\Xi^{E_i}(x)\ge 0\) for every \(x\in X\)
Greater0: \(\sum_{i=0}^n a_i\mu E_i\ge 0\)
Page: 44
Chapter: 12
Section: 2C
Variables: \((X,\Sigma,\mu)\) a measure set,
\(E_0,\dots,E_n\subseteq X\),
\(a_0,\dots, a_n\in\mathbb R\)
Index: (c)
Kind: Lem
Assuming: \(E_0,\dots,E_n\in\Sigma\)

Name: The integral of \(f\)
Definition: \(\sum_{i=0}^ma_i\mu E_i\) where \(f=\sum_{i=0}^ma_i\Xi^{E_i}\).
Page: 44
Chapter: 12
Section: 2E
Variables: \((X,\Sigma,\mu)\) a measure set,
\(f:X\to\mathbb R\) simple
Context: Measure theory
Notation: \(\int f\)
Kind: Def

Name: Integration of simple function
Page: 45
Chapter: 12
Section: 2F
Kind: Prop
Typ: Linear function, respecting pointwise order

Name: A monotone function on \(\mathcal P(X)\)
Definition: A function \(\phi\) such that \(A\subseteq B\subseteq X\) implies \(\phi(A)\le\phi(B)\).
Page: wp
Variables: \(X\) a set

Name: A subadditive function on \(\mathcal P(X)\)
Definition: A function \(\phi\) such that \(\phi(\bigcup A_i)\le \sum \phi (A_i)\) for arbitrary countable set of \((A_i)\).
Page: wp
Variables: \(X\) a set

Smaller0: \(\text{sup}\int f_n<\infty\)
Greater0: \(\text{sup}\{\int g\mid g\text{ simple function and }g\le_{a.e.}\lim f_n\} \)
Page: 45
Chapter: 12
Section: 2I
Variables: \((X,\Sigma,\mu)\) a measured space,
Let \((f_n)_{n\in\mathbb N}:X\to[0,\infty)\) simple,
Let \(f:X\rightharpoondown[0,\infty)\) of conegligible domain
Kind: Lem
Smaller1: \(\lim f_n(x)\) is defined a.e.
Greater1: \(\text{sup}_{n\in\mathbb N}\int f_n\)
Connect Greater0: =
Smaller2: \((f_n)\) non-decreasing

Definition: -\(f\)'s domain is conegligible, and
-there is a non-decreasing sequence \((f_n)\) of non-negative simple functions such that \(\text{sup}_n\int f_n<\infty\) and \(\lim_nf_n(x)=f(x)\) a.e.
Page: 45
Chapter: 12
Section: 2H, 2J
Variables: \((X,\Sigma,\mu)\) a measured space
Notation: \(U\)
Index: , a
Kind: Lem
Definition2: there is a conegligible measurable set \(E\subseteq\text{dom }f\) such that:
-\(f\upharpoonright E\) is measurable,
-\(\mu\{x\in E\mid f(x)\ge \epsilon\}<\infty\) for each \(\epsilon>0\) and
-\(\text{sup}\{\int g\mid g\text{ simple and }g\le_{a.e.}f\}<\infty\)
Conjdef: The set of functions \(f:X\rightharpoondown[0,\infty)\) such that, equivalently

Smaller0: \(f\in U\)
Greater0: \(h\in U\)
Page: 45
Chapter: 12
Section: 2J
Variables: \((X,\Sigma,\mu)\) a measured space,
\(f,h:X\rightharpoondown[0,\infty)\) of conegligible domain,
Index: b
Kind: Lem
Smaller1: \(h\le_{a.e.}f\)
Smaller2: there is a conegligible \(F\subseteq X\) such that \(h\upharpoonright F\) is measurable

Definition: \(\text{sup}\left\{\int g\mid g\text{ simple and }g\le_{a.e.}f\right\}\)
Page: 46
Chapter: 12
Section: 2K, 2L
Variables: \(f:X\rightharpoondown[0,\infty)\) of conegligible domain
Notation: \(\int f\)
Kind: Def, Lem
Typ: Element of \([0,\infty]\)
Assuming: \(f\in U\)

Page: 46
Section: 2K, 2L
Chapter: 12
Notation: \(\int\)
Name: Integration operator
Kind: Def, Lem
Monotonic: Yes
Linear: linear function

Smaller0: \(f^0=_{a.e.}f^1\)
Greater0: \(\int f^0=\int f^1\)
Page: 46
Chapter: 12
Section: 2L
Variables: \(f^0,f^1:X\rightharpoondown[0,\infty)\)
Kind: Lem
Assuming: \(f^0,f^1\in U\)

Name: An integrable real-valued function
Definition: \(f,g\in U\)
Page: 47, 63, 65
Chapter: 12, 13
Section: 2, 3, 3
Variables: \((X,\Sigma,\mu)\) a measure space
Index: M, A, J
Kind: Def, Prop
Definition2: \(\int f\) or \(\int g\) finite
Conjdef: A function of the form \(f-g\) with
Definition3: or, equivalently, \(\underline\int f=\overline\int f\in\mathbb R\)
Name2: A real-valued function integrable over \(X\)
Name3: A real-valued function \(\mu\)-integrable over \(X\)

Name: The integral of \(f\)
Definition: \(\int g-\int h\) where \(f=g-h\) and \(g,h\in U\)
Page: 47
Chapter: 12
Section: 2M
Variables: \((X,\Sigma,\mu)\) a measure space,
\(f:X\to\mathbb R\)
Context: Measure theory
Notation: \(\int f\), \(\int_Xf\), [$]\int f\dd\mu[/$], [$]\int_Xf\dd\mu[/$], [$]\int f(x)\dd\mu(x)[/$], [$]\int_Xf(x)\dd\mu(x)[/$],
Index: (b)
Kind: Rem
Typ: A monotone linear function

Name: A Lebesgue integrable function
Definition: A \((\mathbb R^r,\mathcal L,\mu)\)-integrable function where \(\mathcal L\) is the set of Lebesgue measurable space and \(\mu\) is the Lebesgue measure
Page: 47
Chapter: 12
Section: 2M
Index: (b)
Kind: Def

Name: The Lebesgue integral of \(f\)
Definition: The integral of \(f\) according to the Lesbegue measure
Page: 47
Chapter: 12
Section: 2M
Variables: \(f:\mathbb R^r\to\mathbb R\) a Lebesgue integrable function
Index: (b)
Kind: Def

Name: A \((\mu)\)-virtually measurable function
Definition: A partial function defined on \(X\) whose domain is conegligible
Page: 48
Chapter: 12
Section: 2Q
Variables: \((X,\Sigma,\mu)\) a measure space,
Intuition: negligible set which we ignore

Smaller0: \(f_n\) is non-decreasing
Greater0: \(\lim_nf_n\) is integrable
Page: 52
Chapter: 12
Section: 3A
Variables: \((X,\Sigma,\mu)\) a measure space,
\((f_n)\) a sequence of real-valued functions, integrable over \(X\),
Smaller1: \(\text{sup}_n\int f_n<\infty\)
Name: B Levi's Theorem
Greater1: \(\int\lim_n f_n=\lim_n\int f_n\)

Smaller0: \(\int\lim\text{inf}f_n\)
Greater0: \(\lim\text{inf}\int f_n\)
Page: 53
Chapter: 12
Section: 3B
Variables: \((X,\Sigma,\mu)\) a measure space,
\((f_n):X\to\mathbb R\)
Smaller to greater: \(\le\)
Assuming: \((f_n):X\to\mathbb R^{\ge0}\) integrable
Name: Fatou's lemma
Greater to smaller: \(\ge\)
Hide sides: yes

Smaller0: there exists \(g\) integrable such that \(|f_n|\le_{a.e.}g\) for every \(n\)
Greater0: \(\lim_{n} f_n\) is integrable
Page: 53, 64
Chapter: 12, 13
Section: 3C, 3F
Variables: \((X,\Sigma,\mu)\) a measure space,
\((f_n):X\to\mathbb C\)
Hide middle: yes
Smaller1: \(\lim_{n}\text{inf}\int f_n<\infty\)
Name: Lebesgue's Dominated Convergence Theorem
Greater1: \(\lim_{n}\int f_n=\int\lim_{n} f_n\)
Smaller2: \(f_n\) integrable
Smaller3: \(f_n\) converges pointwise

Smaller0: [$]\int_X f(x,t)\dd x[/$] is defined for every \(t\in]a,b[\)
Greater0: [$]\partialderivative{\int_X f(x,t)\dd x}{t}[/$]
Page: 53, 64
Chapter: 12, 13
Section: 3D, 3H
Variables: \((X,\Sigma,\mu)\) a measure space,
\(f:X\times ]a,b[\to\mathbb C\)
Kind: Cor, Cor
Smaller1: [$]\partialderivative{f}{t}[/$] is defined everywhere in \(X\times ]a,b[\)
Greater1: [$]\int_X\partialderivative{f(x,t)\dd x}{t}[/$]
Connect Greater0: =
Smaller2: there exists \(g:X\to[0,\infty[\) integrable such that [$]\abs{\partialderivative{f}{t} (x,t)} \le g(x)[/$] for every \(x\in X\) and \(t\in]a,b[\)
Greater2: they both exists
Connect Greater1: and

Definition: The set of \(E\in\Sigma\) such that \(E\subseteq H\)
Page: 56
Chapter: 13
Section: 1
Variables: \((X,\Sigma,\mu)\) a measure space,
\(H\in \Sigma\)
Notation: \(\Sigma_H\)

Definition: The restriction of \(\mu\) to \(\Sigma_H\)
Page: 56
Chapter: 13
Section: 1
Variables: \((X,\Sigma,\mu)\) a measure space,
\(H\in \Sigma\)
Notation: \(\mu_H\)

Name: The subspace measure on \(H\)
Page: 56
Chapter: 13
Section: 1
Variables: \((X,\Sigma,\mu)\) a measure space,
\(H\in \Sigma\)
Notation: \((H,\Sigma_H,\mu_H)\)
Index: A, B
Kind: Prop, Def
Typ: A measure space

Name: The Lebesgue measure on \(H\)
Definition: \(\mu_H\), for \(\mu\) the Lebesgue measure
Page: 56
Chapter: 13
Section: 1
Variables: \(H\subseteq\mathbb R^n\) Lebesgue measurable
Index: B
Kind: Def

Smaller0: \(A\) is \(\mu_H\)-negligeable
Greater0: \(A\) is \(\mu\)-negligeable
Page: 56
Chapter: 13
Section: 1
Variables: \((X,\Sigma,\mu)\) a measure space,
\(H\in \Sigma\),
\(A\subseteq H\)
Index: C
Kind: Lem
Smaller to greater: iff

Smaller0: the subspace measure on \(G\) of the subspace measure on \(H\)
Greater0: the subspace measure on \(G\)
Page: 56
Chapter: 13
Section: 1
Variables: \((X,\Sigma,\mu)\) a measure space,
\(G\subseteq H\in \Sigma\),
Index: C
Kind: Lem
Smaller to greater: equals

Definition: [$]\int(f\upharpoonright H)\dd\mu_H[/$]
Page: 56
Chapter: 13
Section: 1
Variables: \((X,\Sigma,\mu)\) a measure space,
\(H\in \Sigma\),
\(f:X\rightharpoondown\mathbb R\)
Notation: \(\int_Hf\)
Index: D
Kind: Def
Notation2: [$]\int_Hf(x)\mu(\dd x)[/$]

Definition: [$]\defFun{H\cap \text{dom}\, f}Y{x}{f(x)}[/$]
Page: 56
Chapter: 13
Section: 1
Variables: \((X,\Sigma,\mu)\) a measure space,
\(H\in\Sigma\),
\(Y\) a set
\(f:X\rightharpoondown Y\)
Notation: \(f\upharpoonright H\)
Index: D
Kind: Def

Smaller0: \(f\) is integrable
Greater0: \(g\) is integrable
Page: 57
Chapter: 13
Section: 1X
Variables: \((X,\Sigma,\mu)\) a measure space,
\(H\in \Sigma\),
\(f:H\rightharpoondown\mathbb R\),
\(g:X\rightharpoondown\mathbb R\) being \(H\ni x\mapsto f(x)\), \(H\not\ni x\mapsto 0\)
Index: E
Kind: Prop
Smaller to greater: iff

Smaller0: [$]\int f\dd\mu_H[/$]
Greater0: [$]\int f\times\Xi^H\dd\mu[/$]
Page: 57
Chapter: 13
Section: 1X
Variables: \((X,\Sigma,\mu)\) a measure space,
\(H\in\Sigma\),
\(f:H\rightharpoondown\mathbb R\)
Index: E
Kind: Prop
Smaller to greater: =

Smaller0: [$]\int f_H\dd \mu_H[/$]
Greater0: [$]\int f\times_{\mathcal X}H\dd \mu[/$]
Page: 57
Chapter: 13
Section: 1F
Variables: \((X,\Sigma,\mu)\) a measure space,
\(H\in \Sigma\),
\(f:X\rightharpoondown\mathbb R\),
Index: (a)
Kind: Cor
Smaller to greater: =
Assuming: \(f\) defined a.e.

Smaller0: \(f\upharpoonright H\) is \(\mu_H\)-integrable
Greater0: \(f\times_{\mathcal X}H\) is \(\mu\)-integrable
Page: 57, 64
Chapter: 13
Section: 1X, 3D
Variables: \((X,\Sigma,\mu)\) a measure space,
\(H\in \Sigma\),
\(f:X\rightharpoondown\mathbb C\)
Index: F(a), c
Kind: Cor, Def
Smaller to greater: iff
Assuming: \(f\) defined a.e.

Smaller0: \(f=0\) a.e.
Greater0: \(\int_H f=0\) for every \(H\in\Sigma\)
Page: 57
Chapter: 13
Section: 1X
Variables: \((X,\Sigma,\mu)\) a measure space,
\(H\in \Sigma\),
\(f:X\rightharpoondown\mathbb R\),
Index: F(c)
Kind: Cor
Smaller to greater: iff
Assuming: \(f\) integrable

Smaller0: \(f\ge0\) a.e.
Greater0: \(\int_Hf\ge0\) for every \(H\in\Sigma\)
Page: 57
Chapter: 13
Section: 1X
Variables: \((X,\Sigma,\mu)\) a measure space,
\(H\in \Sigma\),
\(f:X\rightharpoondown\mathbb R\),
Index: F(b)
Kind: Cor
Smaller to greater: iff
Assuming: \(f\) integrable

Smaller0: \(f\) is \(\mu_H\)-integrable
Greater0: \(f\) is \(\mu\)-integrable
Page: 57
Chapter: 13
Section: 1X
Variables: \((X,\Sigma,\mu)\) a measure space,
\(H\in \Sigma\),
\(f:X\rightharpoondown\mathbb R\),
Index: G
Kind: Cor
Smaller to greater: iff
Assuming: \(H\) conegligible

Smaller0: [$]\int f_H\dd\mu_H[/$]
Greater0: \(\int f\mu\)
Page: 57
Chapter: 13
Section: 1X
Variables: \((X,\Sigma,\mu)\) a measure space,
\(H\in \Sigma\),
\(f:X\rightharpoondown\mathbb R\),
Index: G
Kind: Cor
Smaller to greater: =
Assuming: \(H\) conegligible

Smaller0: \(\int_H f\ge\int_Hg\) for every \(H\in\Sigma\)
Greater0: \(f\ge g\) a.e.
Page: 57
Chapter: 13
Section: 1X
Variables: \((X,\Sigma,\mu)\) a measure space,
\(f,g:X\rightharpoondown\mathbb R\) \(\mu\)-integrable
Index: H(a)
Kind: Cor
Smaller to greater: iff
Extra: cor consider only the implication

Smaller0: \(\int_H f=\int_Hg\) for every \(H\in\Sigma\)
Greater0: \(f=g\) a.e.
Page: 57
Chapter: 13
Section: 1X
Variables: \((X,\Sigma,\mu)\) a measure space,
\(f,g:X\rightharpoondown\mathbb R\) \(\mu\)-integrable
Index: F(b)
Kind: Cor
Smaller to greater: iff
Extra: cor consider only the implication

Name: The outer measure defined from \(\mu\)
Definition: \(\text{inf}\{\mu E:A \subseteq E\in\Sigma\}\)
Page: 59, 60, 65
Chapter: 13
Section: 2, 3J
Variables: \((X,\Sigma,\mu)\) a measure space,
Notation: \(\mu^*\)
Index: A, B
Kind: Prop, Def
Definition2: \(\overline\int_{\mathcal X}A\)
Conjdef: The function \(\mathcal PX\to [0,\infty]\), sending \(A\) to

Question: The number of \(E\in\Sigma\) such that \(\mu^*A=\mu E\) is
Answer: at least 1
Page: 58, 59
Chapter: 13
Section: 2
Variables: \((X,\Sigma,\mu)\) a measure space,
\(A\subseteq X\)
Index: A (a)
Kind: Prop,
Construction: \(\bigcap E_n\), with \(E_n\in\Sigma\),\(\mu E_n\le\mu^* A+\frac 1n\)

Smaller0: \(E\in\Sigma\)
Greater0: equals \(\mu E\)
Page: 58, 59
Chapter: 13
Section: 2
Variables: \((X,\Sigma,\mu)\) a measure space,
\(E\subseteq X\)
\(\mu^*:\mathcal P X\to[0,\infty]\) the outer measure defined from \(\mu\)
Index: A (c)
Kind: Prop,
Prefix Greater: \(\mu^*E\)

Smaller0: \(\mu^*(\bigcup _n A_n)\)
Greater0: \(\lim_n\mu^*A_n\)
Page: 58, 59
Chapter: 13
Section: 2
Variables: \((X,\Sigma,\mu)\) a measure space,
\((A_n)\subseteq X\)
Index: A (e)
Kind: Prop,
Smaller to greater: =
Assuming: \(A_n\) non-decreasing

Question: Examples of \(\theta\) such that \(\mu^*\ne\theta\)
Answer: \(X=\{a,b\}\), \(\theta\) being 0 on \(\emptyset\), \(1\) on \(X\) and \(\frac 12\) everywhere else.
\(\theta(\{a,b\})=\frac12<1=\frac12+\frac12=\theta(\{a\})+\theta(\{b\})=\theta(\{a,b\}\cap \{b\})+\theta(\{a,b\}\setminus\{b\})\), so singletons are not in the domain of \(\mu\).
\(\mu^*(\{a\})=\mu(\{a,b\})=1\) (since \(\{a,b\}\) is the only measurable superset of \(\{a\}\)), while \(\theta(\{a\})=\frac12\)
Page: 60
Chapter: 13
Section: 2
Variables: \((X,\Sigma)\) a measurable space,
\(\theta\) an outer measure
\(\mu\) the measure obtained by Carathéodory's method
Index: B
Kind: Rem

Definition: \(x\mapsto\lim\text{inf}_{n\to\infty}f_n(x)\) defined on the subset of \(\bigcup_{n\in\mathbb N}\bigcap_{m\ge n}\text{dom }f_n\) where the limit exists.
Page: 39
Chapter: 12
Section: 1F
Variables: \((X,\Sigma)\) a measurable space,
\((f_n)_{n\in\mathbb N}:X\rightharpoondown\mathbb R\) measurable,
Notation: \((\lim\text{inf} _{n\to\infty}f_n)\)
Index: (d) (e)
Kind: Theo
DefType: \(\Sigma\)-measurable

Smaller0: \(\sum_{i=0}^ma_i{} _\chi{E_i}=\sum_{i=0}^nb_i{}_\chi{F_i}\)
Greater0: \(\sum_{i=0}^m a_i\mu E_i=\sum_{i=0}^n a_i\mu F_i\)
Page: 45
Chapter: 12
Section: 2D
Variables: \((X,\Sigma,\mu)\) a measure set,
\(E_0,\dots,E_m, F_0,\dots,F_n\in\Sigma\) of finite measure,
\(a_0,\dots, a_m,b_0,\dots,b_m\in\mathbb R\)
Kind: Cor
Counterexample: \((\{x,y\}, \mathcal P\{x,y\},\cdot\mapsto 1)\), \(E_1=\{a\}\), \(E_2=\{b\}\), \(F_1=\{a,b\}\), \(a_1=a_2=b_1=1\)

Smaller0: \((f_i)\) non-decreasing pointwise
Greater0: \(\int f\le\text{sup}\int f_n\)
Page: 45
Chapter: 12
Section: 2G
Variables: \((X,\Sigma,\mu)\) a measure set,
\(f,(f_i)_{i\in\mathbb N}:X\to\mathbb R\)
Kind: Lem
Assuming: the \(f\) and \(f_i\)'s are simple
Smaller1: \(f\le_{\text{a.e.}}\text{sup} f_n\)
Connect Smaller0: and

Name: The \(\sigma\)-algebra of subsets generated by \(A\)
Definition: The least \(\sigma\)-algebra of subsets containing \(A\).
Page: 13
Chapter: 1
Section: 11G
Variables: \(X\) a set
\(A\subseteq \mathcal PX\) nonempty
Notation: \(\Sigma_A\)
Kind: Ex
Example: \(\Sigma_A=\{\emptyset,X\}\)
Example2: \(\Sigma_A\), is the set of countable or co-countable sets
Applied to: \(A=\emptyset\)
Applied to2: \(A=\{\{a\}\mid a\in X\}\)

Smaller0: \(E\) is Borel
Greater0: \(\Sigma_{\bigcap_{i}\text{dom}f_i}\)
Page: 41
Chapter: 12
Section: 1K
Variables: \((X,\Sigma)\) a measurable space,
\(f_1,\dots,f_r: X\rightharpoondown\mathbb R\)
Let \(E\subseteq\mathbb R^r\).
Index: (a)
Kind: Prop
Hide middle: yes
Prefix Greater: \(\vec f^{-1}[E]\) belongs to
Assuming: the \(f_i\)'s measurable

Smaller0: \(\mu^*(A)\)
Greater0: \(\mu^*(A\cap F)\)
Page: 60
Chapter: 13
Section: 2A
Variables: \((X,\Sigma,\mu)\) a measure space,
\(A,F\subseteq X\),
Smaller to greater: =
Assuming: \(F\in\Sigma\)
Greater1: \(\mu^*(A\setminus F)\)
Connect Greater0: +

Name: A measurable envelope of \(A\)
Definition: \(\mu(F\cap E)=\mu^*(F\cap A)\) for every \(F\in\Sigma\)
Page: 60
Chapter: 13
Section: 2
Variables: \((X,\Sigma,\mu)\) a measure space,
\(A\subseteq X\),
Index: D, E (a), (b)
Kind: , Lem
Definition2: \(\mu F=0\) whenever \(F\in\Sigma\) and \(F\subseteq E\setminus A\)
Conjdef: A set \(A\subseteq E\in\Sigma\) such that equivalently
Definition3: \(\mu E=\mu^*A\) (assuming \(\mu E<\infty)\)
Name2: A measurable cover of \(A\)

Name: The set of pair of the form \((A,E)\) with \(A\) a set and \(E\) a measurable envelope of \(A\)
Page: 61
Section: 2
Chapter: 13
Index: E(c) (d)
Variables: \((X,\Sigma,\mu)\) a measure space,
Under: sending \((A,E)\) to \((A\cap H,E\cap H)\) for \(H\) measurable
Under2: sending \((A_i,E_i)_{i\in\mathbb N}\) to \((\bigcup_i A_i,\bigcup_i E_i)\)
Under3: product ?

Name: A set of full outer measure
Definition: \(X\) is a measurable envelope of \(A\)
Page: 61
Chapter: 13
Section: 2
Variables: \((X,\Sigma,\mu)\) a measure space,
Index: E(e,f)
Definition2: \(\mu^*(F\cap A)=\mu F\) for each \(F\in\Sigma\)
Conjdef: A set \(A\subseteq X\) such that, equivalently
Definition3: \(\mu F=0\) whenever \(F\in \Sigma\) and \(F\subseteq X\setminus A\)
Name2: A thick set
Definition4: \(\mu^*A=\mu X\) (assuming \(\mu X<\infty\))

Name: A regular outer measure
Definition: An outer measure \(\theta\) which is equal the outer measure obtained from the measure obtained by Carathéodory's method from \(\theta\)
Page: 61
Chapter: 13
Section: 2
Variables: \((X,\Sigma,\mu)\) a measure space,
Index: X
Kind: ex

Definition: \(f_{\mid\{x\mid f(x)\in\mathbb R\}}\)
Page: 63
Chapter: 13
Section: 3
Variables: \((X,\Sigma,\mu)\) a measure space,
\(f:X\to[-\infty,+\infty]\)
Context: Extended reals
Notation: \(\tilde f\)
Index: B

Definition: \(\int\tilde f\)
Page: 63
Chapter: 13
Section: 3
Variables: \((X,\Sigma,\mu)\) a measure space,
\(f:X\to[-\infty,+\infty]\)
Notation: \(\int f\)
Index: B

Name: A measurable complex function
Definition: \(u+iv\) for \(u,v:X\to\mathbb R\) measurable complex function
Page: 64
Chapter: 13
Section: 3
Variables: \((X,\Sigma,\mu)\) a measure space,
Index: D

Name: An integrable complex function
Definition: \(u+iv\) for \(u,v:X\to\mathbb R\) integrable complex function
Page: 64
Chapter: 13
Section: 3
Variables: \((X,\Sigma,\mu)\) a measure space,
Index: D

Definition: \(\int u+i\int v\)
Page: 64
Chapter: 13
Section: 3
Variables: \((X,\Sigma,\mu)\) a measure space,
\(u,v:X\to\mathbb R\)
\(f=u+iv\)
Notation: \(\int f\)
Index: D (b)

Definition: \(\max(a,0)\)
Page: 47
Chapter: 12
Section: 2N
Variables: \(a\in\mathbb R\)
Notation: \(a^+\)

Definition: \(\min(a,0)\)
Page: 47
Chapter: 12
Section: 2N
Variables: \(a\in\mathbb R\)
Notation: \(a^-\)

Definition: [$]\defFun{X}{\mathbb R^{-}}{x}{x^-}[/$]
Page: 47, 141
Chapter: 12, 24
Section: 2N, 1D
Variables: \(f:X\to\mathbb R\)
Notation: \(f^-\)
Definition2: \(f\land 0\)

Definition: [$]\defFun{X}{\mathbb R^{+}}{x}{x^+}[/$]
Page: 47, 141
Chapter: 12, 24
Section: 2N, 1D
Variables: \(f:X\to\mathbb R\)
Notation: \(f^+\)
Definition2: \(f\lor 0\)

Definition: there exists \(\epsilon>0\) such that \(\{x\in\mathrm{dom} f\mid f(x)\ge \epsilon\}\) has infinite measure
Page: 63
Chapter: 13
Section: 3A
Variables: \((X,\Sigma,\mu)\) a measure space,
\(f:X\to[0,\infty)\)
Notation: \(\int f=\infty\)
Definition2: \(\sup\{\int h:h\mathrm{\ simple}, h\le_{a.e.}f\}=\infty\)
Conjdef: Either

Definition: [$]\defFun{f^{-1}(\mathbb R)}{\mathbb R}{x}{f(x)}[/$]
Page: 63
Chapter: 13
Section: 3A
Variables: \((X,\Sigma,\mu)\) a measure space,
\(f:X\to\mathbb R\cup S\) with \(S\) a set
Notation: \(\tilde f\)

Definition: \(\int\tilde f\)
Page: 63
Chapter: 13
Section: 3A
Variables: \((X,\Sigma,\mu)\) a measure space,
\(f:X\to\mathbb R\cup S\) with \(S\) a set
Notation: \(\int f\)
Assuming: \(f^{-1}(S)\) is negligible

Name: A measurable complex function
Definition: A complex function whose real and imaginary parts are measurable
Page: 66
Chapter: 13
Section: 3D
Variables: \((X,\Sigma,\mu)\) a measure space,

Name: An integrable complex function
Definition: whose real and imaginary parts are integrable
Page: 66
Chapter: 13
Section: 3D, 3F
Variables: \((X,\Sigma,\mu)\) a measure space,
Kind: Prop
Definition2: \(\mu\)-virtually measurable, such that [$]\abs{f}[/$] is integrable, and
Conjdef: A complex function \(f\), equivalently

Definition: \(\int\mathcal {R}ef+i\int\mathcal Im f\).
Page: 66
Chapter: 13
Section: 3D
Variables: \((X,\Sigma,\mu)\) a measure space,
\(f:X\to\mathbb C\)
Context: Complex function
Notation: \(\int f\)

Name: Integrable real/complex-valued functions
Page: 66
Section: 3F
Chapter: 13
Variables: \((X,\Sigma,\mu)\) a measure space,
Under: sum
Condition: where it is defined
Under2: multiplication
Condition2: by an arbitrary real/complex
Under3: taking real or imaginary part

Name: \(\overline\int f\)
Definition: The infimum of \(\int g\) for \(g\le_{a.e.}g\), where \(\int g\) is defined (potentially \(+\infty\))
Page: 65
Chapter: 13
Section: 3F
Variables: \((X,\Sigma,\mu)\) a measure space,
\(f:X\to\mathbb R\) defined a.e.

Name: \(\underline\int f\)
Definition: The supremum of \(\int g\) for \(g\ge_{a.e.}g\), where \(\int g\) is defined (potentially \(-\infty\))
Page: 65
Chapter: 13
Section: 3F
Variables: \((X,\Sigma,\mu)\) a measure space,
\(f:X\to\mathbb R\) defined a.e.

Smaller0: \(\overline\int f\) is finite
Greater0: \(g\) integrable
Page: 65
Chapter: 13
Section: 3F
Variables: \((X,\Sigma,\mu)\) a measure space,
\(f:X\to\mathbb R\) defined a.e.
Prefix Greater: there is \(g:X\to \mathbb R\) defined a.e. with
Greater1: \(f\le_{a.e.}g\)
Greatest: the set of \(x\in\text{dom} f\cap\text{dom}g\) such that \(g(x)\le f(x)+g_0(x)\) has full outer measure, for every \(g_0:X\to]0,\infty[\) measurable
Greater2: \(\int g=\overline\int f\)

Smaller0: \(\underline\int f\)
Greater0: \(\overline\int f\)
Page: 65
Chapter: 13
Section: 3J
Variables: \((X,\Sigma,\mu)\) a measure space,
\(f,g:X\to\mathbb R\) defined a.e.
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Smaller0: \(\overline\int f+g\)
Greater0: \(\overline\int f+\overline\int g\)
Page: 65
Chapter: 13
Section: 3J
Variables: \((X,\Sigma,\mu)\) a measure space,
\(f,g:X\to\mathbb R\) defined a.e.
Smaller to greater: \(\le\)
Assuming: \(\infty\) and \(-\infty\) are not added
Greater to smaller: \(\ge\)

Smaller0: \(\underline\int f+\underline\int g\)
Greater0: \(\underline\int f+g\)
Page: 65
Chapter: 13
Section: 3J
Variables: \((X,\Sigma,\mu)\) a measure space,
\(f,g:X\to\mathbb R\) defined a.e.
Smaller to greater: \(\le\)
Assuming: \(\infty\) and \(-\infty\) are not added
Greater to smaller: \(\ge\)

Smaller0: \(-\overline\int f\)
Greater0: \(\underline\int -f\)
Page: 65
Chapter: 13
Section: 3J
Variables: \((X,\Sigma,\mu)\) a measure space,
\(f,g:X\to\mathbb R\) defined a.e.
Smaller to greater: =

Smaller0: \(c\ge0\)
Greater0: \(c\overline\int f\)
Page: 65
Chapter: 13
Section: 3J
Variables: \((X,\Sigma,\mu)\) a measure space,
\(f,g:X\to\mathbb R\) defined a.e.
Prefix Greater: \(\overline\int cf\) equals

Smaller0: \(c\ge0\)
Greater0: \(c\underline\int f\)
Page: 65
Chapter: 13
Section: 3J
Variables: \((X,\Sigma,\mu)\) a measure space,
\(f,g:X\to\mathbb R\) defined a.e.
Prefix Greater: \(\underline\int cf\) equals