Title: Theory / Unimportant / Math / Measure theory - Fremlin{ / 24 Function spaces=Semi finite, localizable
Tags: fromImplication hasSeenConjDef closed fromMath mathToSee mid1516974248815 toBinaryRelation toUnary

Definition: The space of functions, equivalent a.e. to a \(\Sigma\)-measurable real-valued function
Page: 138, 139
Chapter: 24
Section: 1A, 1B
Variables: \((X,\Sigma,\mu)\) a measure space
Notation: \(\mathcal L^0\)
Index: , k
Definition2: The space of real-valued, virtually-measurable, functions \(f\) defined on a conegligible susbet of \(X\).
Typ: Real vector space
Notation2: \(\mathcal L^0(\mu)\)

Name: The class of real function measurable with domain \(\omega\), \(\mathcal L^0\)
Page: 139
Section: 1A
Chapter: 24
Variables: \((\omega,\Sigma,\mu)\) a measure space
Under: addition and multiplication
Condition: finite
Under2: composition with a Borel measurable set
Under3: sup, inf, lim, lim sup, lim inf, assuming that they are defined almost everywhere
Under4: product/sum by a constant

Definition: The quotient of \(\mathcal L^0\le_{\text{a.e.}}\) by \(=_{\text{a.e.}}\), with the order \(\le\)
Page: 139, 140
Chapter: 24
Section: 1C, 1D
Variables: \((\omega,\Sigma,\mu)\) a measure space
Notation: \(L^0\)
Index: , d
Notation2: \(L^0(\mu)\)
Property: Riesz space

Definition: The function sending an element of \(\mathcal L^0\) to its equivalence class by \(=_{\text{a.e.}}\)
Page: 139
Section: 1C, D
Chapter: 24
Variables: \((\omega,\Sigma,\mu)\) a measure space,
Notation: \(f\mapsto f^\bullet{}\)
Linear: Linear
Morphism: morphism

Page: 139
Definition: The relation containing \((f^\bullet,g^\bullet)\) iff \(f\le_{\text{a.e.}}g\)
Chapter: 24
Section: 1D
Variables: \((\omega,\Sigma,\mu)\) a measure space,
Notation: \(\le\) on \(L^0\)
Order: Partial linear order relation

Name: A partially ordered linear space
Definition: \(u_i\le v_i\) implies \(\sum u_i\le \sum v_i\), and
Page: 140
Chapter: 24
Section: 1D
Definition2: \(0\le u\in V\) and \(0<\lambda\in\mathbb R\) implies \(\lambda u\ge0\)
Example: \((\mathbb R,\le)\)
Conjdef: A real-vector space \(V\), with a relation \(\le\), such that
Example2: Lexicographical order
Extra: example ?
Example3: \(\vec x\le\vec y\) iff \(\bigwedge_i x_i\le y_i\)
Example4: \(\vec x<\vec y\) iff (\(\bigwedge_i x_i< y_i\) or \(\vec x=\vec y\))

Name: A Riesz space
Definition: partially ordered
Page: 140
Chapter: 24
Section: 1D
Definition2: closed under meet and join.
Example: \(L^p\) with the almost everywhere pointwise partial order
Conjdef: both
Name2: A vector lattice
Example2: \(\mathbb R^2\) with lexicographical order
DefType: a linear space

Definition: \(x\mapsto\sup\{u(x),v(x)\}\)
Page: 140
Chapter: 24
Section: 1E
Variables: \((\omega,\Sigma,\mu)\) a measure space,
\(u,v\in L^0\) or in \(\mathcal L^0\)
Notation: \(u\vee v\)
Index: (d)
Notation2: \(\sup\{u,v\}\)

Definition: \(\inf\{f,g\}\)
Page: 140
Chapter: 24
Section: 1D
Variables: \((\omega,\Sigma,\mu)\) a measure space,
\(f,g\in L^0\) or in \(\mathcal L^0\)
Notation: \(f\wedge g\)
Definition2: \(\frac{f+g-|f-g|}2\)

Definition: [$]x\mapsto\abs{f(x)}[/$]
Page: 141
Chapter: 24
Section: 1D
Variables: \((\omega,\Sigma,\mu)\) a measure space,
\(f\in L^0\) or in \(\mathcal L^0\)
Notation: [$]\abs{f}[/$]
Definition2: \(f^++f^-\)
Definition3: \(f\vee (-f)\)

Smaller0: \(|f^\bullet|\)
Greater0: \(|f|^\bullet\)
Page: 141
Chapter: 24
Section: 1D
Variables: \((\omega,\Sigma,\mu)\) a measure space,
\(f\in\mathcal L^0\)
Smaller to greater: =

Smaller0: \(|c||f|\)
Greater0: \(|cf|\)
Page: 141
Chapter: 24
Section: 1D
Variables: \((\omega,\Sigma,\mu)\) a measure space,
\(f\in L^0\) or in \(\mathcal L^0\),
\(c\in\mathbb R\)
Smaller to greater: =

Smaller0: \(f\vee g\)
Greater0: \(\frac{f+g+|f-g|}2\)
Page: 141
Chapter: 24
Section: 1D
Variables: \((\omega,\Sigma,\mu)\) a measure space,
\(f,g\in L^0\) or in \(\mathcal L^0\)
Smaller to greater: =
Context: Real analysis

Smaller0: \(|f+g|\)
Greater0: \(|f|+|g|\)
Page: 141
Chapter: 24
Section: 1D
Variables: \((\omega,\Sigma,\mu)\) a measure space,
\(f,g\in L^0\) or in \(\mathcal L^0\)
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Question: \(f^+-f^-\)
Answer: equals \(f\)
Page: 141
Chapter: 24
Section: 1D
Variables: \((\omega,\Sigma,\mu)\) a measure space,
\(f,g\in L^0\) or in \(\mathcal L^0\)
Index: f

Question: \(f^+\wedge f^-\)
Answer: equals \(0\)
Page: 141
Chapter: 24
Section: 1D
Variables: \((\omega,\Sigma,\mu)\) a measure space,
\(f,g\in L^0\) or in \(\mathcal L^0\)
Index: f

Name: An Archimedean Riesz space
Definition: A Riesz space \(U\) such that whenever \(0<u\in U\), and \(v\in U\), there is \(n\in\mathbb N\) such that \(nu\not\le v\)
Page: 141
Chapter: 24
Section: 1F
Index: a
Example: On a compact topological space \(X\), the space of function with pointwise partial order
Counterexample: \(\mathbb R^2\) with lexicographical order

Name: A Dedekind \(\sigma\)-complete Riesz space
Definition: every non-empty countable set \(A\subseteq U\) which is bounded above has a least upper bound in \(U\)
Page: 141
Chapter: 24
Section: 1F
Index: b
Conjdef: A Riesz space \(U\) such that
Name2: A Dedekind \(\sigma\)-order-complete Riesz space
Name3: A \(\sigma\)-complete Riesz space

Name: A Dedekind complete Riesz space
Definition: every non-empty set \(A\subseteq U\) bounded above in \(U\) has a least upper bound in \(U\)
Page: 141
Chapter: 24
Section: 1F
Index: c
Name2: An order complete space
DefType: A Riesz space \(U\)
Name3: a complete space