Uses of characterization of $sigma$-finiteness The 2019 Stack Overflow Developer Survey Results Are InFact about measurable functions defined on $sigma$-finite measure spaces.Non-measurable set in product $sigma$-algebra s.t. every section is measurable.Connection between separable measure spaces and $sigma$-finite measure spacesIncomplete measure space that is not sigma-finiteHow does one determine the $sigma$-algebra of $mu^*$-measurable subsets for the following $mu^*$?Show that the area set is measurableNon sigma-finite measure defined by integral (example)measurability of functions; sub sigma field and completionPullback probability measure : how to pullback the finite uniform probability?How to show that $mathscr F$ is a Sigma Algebra?
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Uses of characterization of $sigma$-finiteness
The 2019 Stack Overflow Developer Survey Results Are InFact about measurable functions defined on $sigma$-finite measure spaces.Non-measurable set in product $sigma$-algebra s.t. every section is measurable.Connection between separable measure spaces and $sigma$-finite measure spacesIncomplete measure space that is not sigma-finiteHow does one determine the $sigma$-algebra of $mu^*$-measurable subsets for the following $mu^*$?Show that the area set is measurableNon sigma-finite measure defined by integral (example)measurability of functions; sub sigma field and completionPullback probability measure : how to pullback the finite uniform probability?How to show that $mathscr F$ is a Sigma Algebra?
$begingroup$
It's not difficult to prove the following characterization of $sigma$-finite measures: Let $(Omega, mathscr A, mu)$ be a measure space. Then,
$$ Omega text is sigmatext-finite iff textthere exists a measurable f > 0 text with int_Omega f d mu < infty.$$
My question: Is this just a common practice problems for students to get familiar with $sigma$-finite spaces or do you know any situations where this fact came in handy?
measure-theory
asked Apr 7 at 18:07
KezerKezer
1,405621
$endgroup$
add a comment |
$begingroup$
It's not difficult to prove the following characterization of $sigma$-finite measures: Let $(Omega, mathscr A, mu)$ be a measure space. Then,
$$ Omega text is sigmatext-finite iff textthere exists a measurable f > 0 text with int_Omega f d mu < infty.$$
My question: Is this just a common practice problems for students to get familiar with $sigma$-finite spaces or do you know any situations where this fact came in handy?
measure-theory
asked Apr 7 at 18:07
KezerKezer
1,405621
$endgroup$
add a comment |
$begingroup$
It's not difficult to prove the following characterization of $sigma$-finite measures: Let $(Omega, mathscr A, mu)$ be a measure space. Then,
$$ Omega text is sigmatext-finite iff textthere exists a measurable f > 0 text with int_Omega f d mu < infty.$$
My question: Is this just a common practice problems for students to get familiar with $sigma$-finite spaces or do you know any situations where this fact came in handy?
measure-theory
asked Apr 7 at 18:07
KezerKezer
1,405621
$endgroup$
It's not difficult to prove the following characterization of $sigma$-finite measures: Let $(Omega, mathscr A, mu)$ be a measure space. Then,
$$ Omega text is sigmatext-finite iff textthere exists a measurable f > 0 text with int_Omega f d mu < infty.$$
My question: Is this just a common practice problems for students to get familiar with $sigma$-finite spaces or do you know any situations where this fact came in handy?
measure-theory
measure-theory
asked Apr 7 at 18:07
KezerKezer
1,405621
asked Apr 7 at 18:07
KezerKezer
1,405621
asked Apr 7 at 18:07
KezerKezer
1,405621
asked Apr 7 at 18:07
KezerKezer
1,405621
asked Apr 7 at 18:07
KezerKezer
1,405621
1,405621
add a comment |
add a comment |
1 Answer
1
active
oldest
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$begingroup$
I think it serves as "motivation": we want to study $sigma$-finite measures so at least we have some non-trivial integrals for positive functions. As integrals are functionals for some function spaces, this shows that in those cases we have at least some non-trivial functionals. It justifies why we often have $sigma$-finiteness as an assumption in theorems.
answered Apr 7 at 18:12
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
$begingroup$
Great answer! Motivation is at least as important (or even more important) than simple applications for me!
$endgroup$
– Kezer
Apr 7 at 18:20
add a comment |
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1 Answer
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1 Answer
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$begingroup$
I think it serves as "motivation": we want to study $sigma$-finite measures so at least we have some non-trivial integrals for positive functions. As integrals are functionals for some function spaces, this shows that in those cases we have at least some non-trivial functionals. It justifies why we often have $sigma$-finiteness as an assumption in theorems.
answered Apr 7 at 18:12
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
$begingroup$
Great answer! Motivation is at least as important (or even more important) than simple applications for me!
$endgroup$
– Kezer
Apr 7 at 18:20
add a comment |
$begingroup$
I think it serves as "motivation": we want to study $sigma$-finite measures so at least we have some non-trivial integrals for positive functions. As integrals are functionals for some function spaces, this shows that in those cases we have at least some non-trivial functionals. It justifies why we often have $sigma$-finiteness as an assumption in theorems.
answered Apr 7 at 18:12
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
$begingroup$
Great answer! Motivation is at least as important (or even more important) than simple applications for me!
$endgroup$
– Kezer
Apr 7 at 18:20
add a comment |
$begingroup$
I think it serves as "motivation": we want to study $sigma$-finite measures so at least we have some non-trivial integrals for positive functions. As integrals are functionals for some function spaces, this shows that in those cases we have at least some non-trivial functionals. It justifies why we often have $sigma$-finiteness as an assumption in theorems.
answered Apr 7 at 18:12
Henno BrandsmaHenno Brandsma
116k349127
$endgroup$
I think it serves as "motivation": we want to study $sigma$-finite measures so at least we have some non-trivial integrals for positive functions. As integrals are functionals for some function spaces, this shows that in those cases we have at least some non-trivial functionals. It justifies why we often have $sigma$-finiteness as an assumption in theorems.
answered Apr 7 at 18:12
Henno BrandsmaHenno Brandsma
116k349127
answered Apr 7 at 18:12
Henno BrandsmaHenno Brandsma
116k349127
answered Apr 7 at 18:12
Henno BrandsmaHenno Brandsma
116k349127
answered Apr 7 at 18:12
Henno BrandsmaHenno Brandsma
116k349127
116k349127
$begingroup$
Great answer! Motivation is at least as important (or even more important) than simple applications for me!
$endgroup$
– Kezer
Apr 7 at 18:20
add a comment |
$begingroup$
Great answer! Motivation is at least as important (or even more important) than simple applications for me!
$endgroup$
– Kezer
Apr 7 at 18:20
$begingroup$
Great answer! Motivation is at least as important (or even more important) than simple applications for me!
$endgroup$
– Kezer
Apr 7 at 18:20
$begingroup$
Great answer! Motivation is at least as important (or even more important) than simple applications for me!
$endgroup$
– Kezer
Apr 7 at 18:20
add a comment |
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