Metric space Cat. I and Cat. 2
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In Munroe1956, Introduction to Measure Theory and Integration, I found this exercise
Let $Omega$ be a complete metric space. Let $mu^*(E)=0$ if $E$ is of Cat. I, $mu^*(E)=1$ if E is of Cat. II. Show that $mu^*$ is an outer measure, and determine the class of measurable sets.
Are the Cat. I and Cat. II connected with the CAT(k) space concept or they mean something else?
general-topology measure-theory
asked Dec 6 '18 at 7:07
PeptideChainPeptideChain
464311
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add a comment |
$begingroup$
In Munroe1956, Introduction to Measure Theory and Integration, I found this exercise
Let $Omega$ be a complete metric space. Let $mu^*(E)=0$ if $E$ is of Cat. I, $mu^*(E)=1$ if E is of Cat. II. Show that $mu^*$ is an outer measure, and determine the class of measurable sets.
Are the Cat. I and Cat. II connected with the CAT(k) space concept or they mean something else?
general-topology measure-theory
asked Dec 6 '18 at 7:07
PeptideChainPeptideChain
464311
$endgroup$
1
$begingroup$
A Cat I set is a countable union of nowhere dense sets. A Cat II set is one which is not of Cat I.
$endgroup$
– Kavi Rama Murthy
Dec 6 '18 at 7:15
add a comment |
$begingroup$
In Munroe1956, Introduction to Measure Theory and Integration, I found this exercise
Let $Omega$ be a complete metric space. Let $mu^*(E)=0$ if $E$ is of Cat. I, $mu^*(E)=1$ if E is of Cat. II. Show that $mu^*$ is an outer measure, and determine the class of measurable sets.
Are the Cat. I and Cat. II connected with the CAT(k) space concept or they mean something else?
general-topology measure-theory
asked Dec 6 '18 at 7:07
PeptideChainPeptideChain
464311
$endgroup$
In Munroe1956, Introduction to Measure Theory and Integration, I found this exercise
Let $Omega$ be a complete metric space. Let $mu^*(E)=0$ if $E$ is of Cat. I, $mu^*(E)=1$ if E is of Cat. II. Show that $mu^*$ is an outer measure, and determine the class of measurable sets.
Are the Cat. I and Cat. II connected with the CAT(k) space concept or they mean something else?
general-topology measure-theory
general-topology measure-theory
asked Dec 6 '18 at 7:07
PeptideChainPeptideChain
464311
asked Dec 6 '18 at 7:07
PeptideChainPeptideChain
464311
asked Dec 6 '18 at 7:07
PeptideChainPeptideChain
464311
asked Dec 6 '18 at 7:07
PeptideChainPeptideChain
464311
asked Dec 6 '18 at 7:07
PeptideChainPeptideChain
464311
464311
1
$begingroup$
A Cat I set is a countable union of nowhere dense sets. A Cat II set is one which is not of Cat I.
$endgroup$
– Kavi Rama Murthy
Dec 6 '18 at 7:15
add a comment |
1
$begingroup$
A Cat I set is a countable union of nowhere dense sets. A Cat II set is one which is not of Cat I.
$endgroup$
– Kavi Rama Murthy
Dec 6 '18 at 7:15
1
1
$begingroup$
A Cat I set is a countable union of nowhere dense sets. A Cat II set is one which is not of Cat I.
$endgroup$
– Kavi Rama Murthy
Dec 6 '18 at 7:15
$begingroup$
A Cat I set is a countable union of nowhere dense sets. A Cat II set is one which is not of Cat I.
$endgroup$
– Kavi Rama Murthy
Dec 6 '18 at 7:15
add a comment |
1 Answer
1
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oldest
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$begingroup$
No, Baire category has nothing to do with $text{CAT}(k)$.
Baire's terminology Category I , Category II is from 1899.
Notation $text{CAT}(k)$ is from 1987.
Alternate terminology from Bourbaki: nowhere dense = rare, Category I = meagre, Category II = nonmeagre.
answered Dec 6 '18 at 12:04
GEdgarGEdgar
62.8k267171
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add a comment |
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$begingroup$
No, Baire category has nothing to do with $text{CAT}(k)$.
Baire's terminology Category I , Category II is from 1899.
Notation $text{CAT}(k)$ is from 1987.
Alternate terminology from Bourbaki: nowhere dense = rare, Category I = meagre, Category II = nonmeagre.
answered Dec 6 '18 at 12:04
GEdgarGEdgar
62.8k267171
$endgroup$
add a comment |
$begingroup$
No, Baire category has nothing to do with $text{CAT}(k)$.
Baire's terminology Category I , Category II is from 1899.
Notation $text{CAT}(k)$ is from 1987.
Alternate terminology from Bourbaki: nowhere dense = rare, Category I = meagre, Category II = nonmeagre.
answered Dec 6 '18 at 12:04
GEdgarGEdgar
62.8k267171
$endgroup$
add a comment |
$begingroup$
No, Baire category has nothing to do with $text{CAT}(k)$.
Baire's terminology Category I , Category II is from 1899.
Notation $text{CAT}(k)$ is from 1987.
Alternate terminology from Bourbaki: nowhere dense = rare, Category I = meagre, Category II = nonmeagre.
answered Dec 6 '18 at 12:04
GEdgarGEdgar
62.8k267171
$endgroup$
No, Baire category has nothing to do with $text{CAT}(k)$.
Baire's terminology Category I , Category II is from 1899.
Notation $text{CAT}(k)$ is from 1987.
Alternate terminology from Bourbaki: nowhere dense = rare, Category I = meagre, Category II = nonmeagre.
answered Dec 6 '18 at 12:04
GEdgarGEdgar
62.8k267171
answered Dec 6 '18 at 12:04
GEdgarGEdgar
62.8k267171
answered Dec 6 '18 at 12:04
GEdgarGEdgar
62.8k267171
answered Dec 6 '18 at 12:04
GEdgarGEdgar
62.8k267171
62.8k267171
add a comment |
add a comment |
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$begingroup$
A Cat I set is a countable union of nowhere dense sets. A Cat II set is one which is not of Cat I.
$endgroup$
– Kavi Rama Murthy
Dec 6 '18 at 7:15