Title: Theory / Unimportant / Math / Measure theory - Fremlin{ / 25 Product measure=semi-finite, sigma finite...
Tags: mid1516974248815 toUnary fromImplication

Definition: The infimum of \(\sum_{n=0}^\infty\mu_1 E^1_n\cdot \mu_2 E^2_n\), with \(A\subseteq\bigcup_n E^1_n\times E^2_n\), \(E^i_n\subseteq X_i\)
Page: 204
Section: 1A B
Chapter: 25
Variables: \((X_i,\Sigma_i,\mu_i)\) measure spaces,
\(A\subseteq X_1\times X_2\)
Notation: \(\theta A\)
Measure: Outer measure

Name: The primitive product measure on \(X_1\times X_2\)
Definition: The measure derived by Carathéodory's method from the outer measure \(\theta\)
Page: 204
Chapter: 25
Section: 1C
Variables: \((X_i,\Sigma_i,\mu_i)\) measure spaces
Notation: \(X_1\hat\otimes X_2\)

Definition: [$]\defFun{\text{dom }\lambda_0}{[0,\infty]}{W}{\text{sup}\{\lambda_0(W\cap (E_1\times E_2))\mid E_i\in\Sigma_1, \mu_i(E_i)<\infty\}}[/$]
with \(\lambda_0\) the primitive product measure, \(E_i\in\Sigma_i\) of finite measure
Page: 206, 207
Section: 1F, G, I
Chapter: 22
Variables: \((X_i,\Sigma_i,\mu_i)\) measure spaces
Notation: \(\lambda\)
Name: The c.l.d. product measure on \(X_1\times X_2\)
Index: Def, Theo
Measurable: Complete measure, locally determined

Smaller0: \(\lambda_0\) is defined on \(H\cap (E_1\times E_2)\) whenever \(\mu E_i<\infty\)
Greater0: \(\lambda_0\) is defined on \(H\)
Page: 206
Chapter: 25
Section: 1H
Variables: \((X_i,\Sigma_i,\mu_i)\) measure spaces,
\(H\subseteq X_1\times X_2\)

Smaller0: \(\Sigma_1\hat\otimes\Sigma_2\)
Greater0: \(\Lambda\)
Page: 207
Chapter: 25
Section: 1I
Variables: \((X_i,\Sigma_i,\mu_i)\) measure spaces,
\(\lambda\) the c.l.d. product measure on \(X_1\times X_2\),
\(\Lambda\) the domain of \(\lambda\)
Index: (a)
Kind: Theo
Smaller to greater: \(\subseteq\)
Greater to smaller: \(\supseteq\)

Smaller0: \(\prod \mu_i E_i\)
Greater0: \(\lambda(\prod E_i)\)
Page: 207
Chapter: 25
Section: 1I
Variables: \((X_i,\Sigma_i,\mu_i)\) measure spaces,
\(H\subseteq X_1\times X_2\),
\(\lambda\) the c.l.d. product measure on \(X_1\times X_2\),
\(\Lambda\) the domain of \(\lambda\),
\(E_i\subseteq \Sigma_i\)
Index: (a)
Kind: Theo
Smaller to greater: =
Assuming: \(\mu E_i<\infty\)

Smaller0: \(W\in\Lambda\)
Greater0: there exists \(V\in\Sigma_1\hat\otimes\Sigma_2\) such that \(V\subseteq W\) and \(\lambda V=\lambda W\)
Page: 207
Chapter: 25
Section: 1I
Variables: \((X_i,\Sigma_i,\mu_i)\) measure spaces,
\(H\subseteq X_1\times X_2\),
\(\lambda\) the c.l.d. product measure on \(X_1\times X_2\),
\(\Lambda\) the domain of \(\lambda\),
\(W\subseteq\prod X_i\)
Index: (b)
Kind: Theo
Context: Measure theory, c.l.d. product

Question: todo
Answer: after section 213DE
Page: 207
Chapter: 25
Section: 1I
Variables: \((X_i,\Sigma_i,\mu_i)\) measure spaces,
\(H\subseteq X_1\times X_2\),
\(\lambda\) the c.l.d. product measure on \(X_1\times X_2\),
\(\Lambda\) the domain of \(\lambda\),
Index: (c)
Kind: Theo

Smaller0: \(W\in\Lambda\)
Greater0: there are \((E_i\in\Sigma_i)_i\) such that \(\forall i.\mu E_i<\infty\) and \(\lambda(W\cap \prod E_i)>0\)
Page: 207
Chapter: 25
Section: 1I
Variables: \((X_i,\Sigma_i,\mu_i)\) measure spaces,
\(H\subseteq X_1\times X_2\),
\(\lambda\) the c.l.d. product measure on \(X_1\times X_2\),
\(\Lambda\) the domain of \(\lambda\),
\(D\subseteq \prod X_i\)
Index: (d)
Kind: Theo
Smaller1: \(\lambda W>0\)
Context: C.l.d. product of measure spaces

Smaller0: \(W\in\Lambda\)
Greater0: for every \(\epsilon>0\), there are \((E_i^j)_{0\le j\le n}\in\Sigma_i\) of finite measures such that \(\lambda(W\Delta\bigcup_j \prod_i E_i)<0\)
Page: 207
Chapter: 25
Section: 1I
Variables: \((X_i,\Sigma_i,\mu_i)\) measure spaces,
\(H\subseteq X_1\times X_2\),
\(\lambda\) the c.l.d. product measure on \(X_1\times X_2\),
\(\Lambda\) the domain of \(\lambda\),
\(D\subseteq \prod X_i\)
Index: (d)
Kind: Theo
Smaller1: \(\lambda W<\infty\)