Hahn-Vitali-Saks theorem: do we need to ask for finite total variations?
$begingroup$
In Yosida's functional analysis (p70), we encounter the Hahn-Vitali-Saks theorem, as also stated on wikipedia:
https://en.wikipedia.org/wiki/Vitali%E2%80%93Hahn%E2%80%93Saks_theorem
In particular, one of the conditions is that for all $n geq 1$, the complex measures $lambda_n$ must have finite total variations $|lambda_n|(S)$.
However, in Rudin's book Real and Complex analysis, in theorem 6.4 (p119) we read that the total variation of a complex measure is always finite (it is even bounded!)
Does this mean that we can drop the condition that the complex measures $lambda_n$ must have finite total variations in the formulation of the Hahn-Vitali-Saks theorem, as this condition is always satisfied? Or am I missing something?
functional-analysis measure-theory total-variation
asked Dec 1 '18 at 14:17
Math_QEDMath_QED
7,58331452
$endgroup$
add a comment |
$begingroup$
In Yosida's functional analysis (p70), we encounter the Hahn-Vitali-Saks theorem, as also stated on wikipedia:
https://en.wikipedia.org/wiki/Vitali%E2%80%93Hahn%E2%80%93Saks_theorem
In particular, one of the conditions is that for all $n geq 1$, the complex measures $lambda_n$ must have finite total variations $|lambda_n|(S)$.
However, in Rudin's book Real and Complex analysis, in theorem 6.4 (p119) we read that the total variation of a complex measure is always finite (it is even bounded!)
Does this mean that we can drop the condition that the complex measures $lambda_n$ must have finite total variations in the formulation of the Hahn-Vitali-Saks theorem, as this condition is always satisfied? Or am I missing something?
functional-analysis measure-theory total-variation
asked Dec 1 '18 at 14:17
Math_QEDMath_QED
7,58331452
$endgroup$
add a comment |
$begingroup$
In Yosida's functional analysis (p70), we encounter the Hahn-Vitali-Saks theorem, as also stated on wikipedia:
https://en.wikipedia.org/wiki/Vitali%E2%80%93Hahn%E2%80%93Saks_theorem
In particular, one of the conditions is that for all $n geq 1$, the complex measures $lambda_n$ must have finite total variations $|lambda_n|(S)$.
However, in Rudin's book Real and Complex analysis, in theorem 6.4 (p119) we read that the total variation of a complex measure is always finite (it is even bounded!)
Does this mean that we can drop the condition that the complex measures $lambda_n$ must have finite total variations in the formulation of the Hahn-Vitali-Saks theorem, as this condition is always satisfied? Or am I missing something?
functional-analysis measure-theory total-variation
asked Dec 1 '18 at 14:17
Math_QEDMath_QED
7,58331452
$endgroup$
In Yosida's functional analysis (p70), we encounter the Hahn-Vitali-Saks theorem, as also stated on wikipedia:
https://en.wikipedia.org/wiki/Vitali%E2%80%93Hahn%E2%80%93Saks_theorem
In particular, one of the conditions is that for all $n geq 1$, the complex measures $lambda_n$ must have finite total variations $|lambda_n|(S)$.
However, in Rudin's book Real and Complex analysis, in theorem 6.4 (p119) we read that the total variation of a complex measure is always finite (it is even bounded!)
Does this mean that we can drop the condition that the complex measures $lambda_n$ must have finite total variations in the formulation of the Hahn-Vitali-Saks theorem, as this condition is always satisfied? Or am I missing something?
functional-analysis measure-theory total-variation
functional-analysis measure-theory total-variation
asked Dec 1 '18 at 14:17
Math_QEDMath_QED
7,58331452
asked Dec 1 '18 at 14:17
Math_QEDMath_QED
7,58331452
edited Dec 1 '18 at 16:30
Math_QED
asked Dec 1 '18 at 14:17
Math_QEDMath_QED
7,58331452
asked Dec 1 '18 at 14:17
Math_QEDMath_QED
7,58331452
asked Dec 1 '18 at 14:17
Math_QEDMath_QED
7,58331452
7,58331452
add a comment |
add a comment |
1 Answer
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active
oldest
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When Rudin defines "complex measure" he requires the measure to be finite everywhere. That excludes many well-behaved measures, including Lebesgue measure.
I think it's also common to use "complex measure" to simply emphasize that the measure is not necessarily positive, but it might be allowed to include infinite measure cases.
answered Dec 1 '18 at 15:10
Martin ArgeramiMartin Argerami
127k1182182
$endgroup$
$begingroup$
Thanks for your answer. Yosida seems to define complex measure as a set function $varphi$ that is $sigma$-additive and such that $|varphi(B)| neq infty$ for all $B$, so I'm not entirely sure if you completely answered the question.
$endgroup$
– Math_QED
Dec 1 '18 at 16:02
$begingroup$
Moreover, the Lebesgue measure has total variation $infty$ so the theorem would exclude it anyway.
$endgroup$
– Math_QED
Dec 1 '18 at 16:04
$begingroup$
That's my point, the theorem excludes Lebesgue measure because it doesn't have finite total variation. Rudin would exclude it by definition. Now, if Yosida defines complex measure the same as Rudin, there is no need to require the hypothesis.
$endgroup$
– Martin Argerami
Dec 1 '18 at 16:14
$begingroup$
I don't know if you have a copy of Yosida at hand or even own the book, but I think he defines complex measures on p35. He uses an alternative definition, which I believe is equivalent with Rudin's. This equivalence is an exercise in Folland's real analysis. It sure is confusing haha.
$endgroup$
– Math_QED
Dec 1 '18 at 16:19
1
$begingroup$
True, but you can apply it to the real and imaginary parts of your measure.
$endgroup$
– Martin Argerami
Dec 1 '18 at 17:20
|
show 5 more comments
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
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active
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active
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votes
$begingroup$
When Rudin defines "complex measure" he requires the measure to be finite everywhere. That excludes many well-behaved measures, including Lebesgue measure.
I think it's also common to use "complex measure" to simply emphasize that the measure is not necessarily positive, but it might be allowed to include infinite measure cases.
answered Dec 1 '18 at 15:10
Martin ArgeramiMartin Argerami
127k1182182
$endgroup$
$begingroup$
Thanks for your answer. Yosida seems to define complex measure as a set function $varphi$ that is $sigma$-additive and such that $|varphi(B)| neq infty$ for all $B$, so I'm not entirely sure if you completely answered the question.
$endgroup$
– Math_QED
Dec 1 '18 at 16:02
$begingroup$
Moreover, the Lebesgue measure has total variation $infty$ so the theorem would exclude it anyway.
$endgroup$
– Math_QED
Dec 1 '18 at 16:04
$begingroup$
That's my point, the theorem excludes Lebesgue measure because it doesn't have finite total variation. Rudin would exclude it by definition. Now, if Yosida defines complex measure the same as Rudin, there is no need to require the hypothesis.
$endgroup$
– Martin Argerami
Dec 1 '18 at 16:14
$begingroup$
I don't know if you have a copy of Yosida at hand or even own the book, but I think he defines complex measures on p35. He uses an alternative definition, which I believe is equivalent with Rudin's. This equivalence is an exercise in Folland's real analysis. It sure is confusing haha.
$endgroup$
– Math_QED
Dec 1 '18 at 16:19
1
$begingroup$
True, but you can apply it to the real and imaginary parts of your measure.
$endgroup$
– Martin Argerami
Dec 1 '18 at 17:20
|
show 5 more comments
$begingroup$
When Rudin defines "complex measure" he requires the measure to be finite everywhere. That excludes many well-behaved measures, including Lebesgue measure.
I think it's also common to use "complex measure" to simply emphasize that the measure is not necessarily positive, but it might be allowed to include infinite measure cases.
answered Dec 1 '18 at 15:10
Martin ArgeramiMartin Argerami
127k1182182
$endgroup$
$begingroup$
Thanks for your answer. Yosida seems to define complex measure as a set function $varphi$ that is $sigma$-additive and such that $|varphi(B)| neq infty$ for all $B$, so I'm not entirely sure if you completely answered the question.
$endgroup$
– Math_QED
Dec 1 '18 at 16:02
$begingroup$
Moreover, the Lebesgue measure has total variation $infty$ so the theorem would exclude it anyway.
$endgroup$
– Math_QED
Dec 1 '18 at 16:04
$begingroup$
That's my point, the theorem excludes Lebesgue measure because it doesn't have finite total variation. Rudin would exclude it by definition. Now, if Yosida defines complex measure the same as Rudin, there is no need to require the hypothesis.
$endgroup$
– Martin Argerami
Dec 1 '18 at 16:14
$begingroup$
I don't know if you have a copy of Yosida at hand or even own the book, but I think he defines complex measures on p35. He uses an alternative definition, which I believe is equivalent with Rudin's. This equivalence is an exercise in Folland's real analysis. It sure is confusing haha.
$endgroup$
– Math_QED
Dec 1 '18 at 16:19
1
$begingroup$
True, but you can apply it to the real and imaginary parts of your measure.
$endgroup$
– Martin Argerami
Dec 1 '18 at 17:20
|
show 5 more comments
$begingroup$
When Rudin defines "complex measure" he requires the measure to be finite everywhere. That excludes many well-behaved measures, including Lebesgue measure.
I think it's also common to use "complex measure" to simply emphasize that the measure is not necessarily positive, but it might be allowed to include infinite measure cases.
answered Dec 1 '18 at 15:10
Martin ArgeramiMartin Argerami
127k1182182
$endgroup$
When Rudin defines "complex measure" he requires the measure to be finite everywhere. That excludes many well-behaved measures, including Lebesgue measure.
I think it's also common to use "complex measure" to simply emphasize that the measure is not necessarily positive, but it might be allowed to include infinite measure cases.
answered Dec 1 '18 at 15:10
Martin ArgeramiMartin Argerami
127k1182182
answered Dec 1 '18 at 15:10
Martin ArgeramiMartin Argerami
127k1182182
answered Dec 1 '18 at 15:10
Martin ArgeramiMartin Argerami
127k1182182
answered Dec 1 '18 at 15:10
Martin ArgeramiMartin Argerami
127k1182182
127k1182182
$begingroup$
Thanks for your answer. Yosida seems to define complex measure as a set function $varphi$ that is $sigma$-additive and such that $|varphi(B)| neq infty$ for all $B$, so I'm not entirely sure if you completely answered the question.
$endgroup$
– Math_QED
Dec 1 '18 at 16:02
$begingroup$
Moreover, the Lebesgue measure has total variation $infty$ so the theorem would exclude it anyway.
$endgroup$
– Math_QED
Dec 1 '18 at 16:04
$begingroup$
That's my point, the theorem excludes Lebesgue measure because it doesn't have finite total variation. Rudin would exclude it by definition. Now, if Yosida defines complex measure the same as Rudin, there is no need to require the hypothesis.
$endgroup$
– Martin Argerami
Dec 1 '18 at 16:14
$begingroup$
I don't know if you have a copy of Yosida at hand or even own the book, but I think he defines complex measures on p35. He uses an alternative definition, which I believe is equivalent with Rudin's. This equivalence is an exercise in Folland's real analysis. It sure is confusing haha.
$endgroup$
– Math_QED
Dec 1 '18 at 16:19
1
$begingroup$
True, but you can apply it to the real and imaginary parts of your measure.
$endgroup$
– Martin Argerami
Dec 1 '18 at 17:20
|
show 5 more comments
$begingroup$
Thanks for your answer. Yosida seems to define complex measure as a set function $varphi$ that is $sigma$-additive and such that $|varphi(B)| neq infty$ for all $B$, so I'm not entirely sure if you completely answered the question.
$endgroup$
– Math_QED
Dec 1 '18 at 16:02
$begingroup$
Moreover, the Lebesgue measure has total variation $infty$ so the theorem would exclude it anyway.
$endgroup$
– Math_QED
Dec 1 '18 at 16:04
$begingroup$
That's my point, the theorem excludes Lebesgue measure because it doesn't have finite total variation. Rudin would exclude it by definition. Now, if Yosida defines complex measure the same as Rudin, there is no need to require the hypothesis.
$endgroup$
– Martin Argerami
Dec 1 '18 at 16:14
$begingroup$
I don't know if you have a copy of Yosida at hand or even own the book, but I think he defines complex measures on p35. He uses an alternative definition, which I believe is equivalent with Rudin's. This equivalence is an exercise in Folland's real analysis. It sure is confusing haha.
$endgroup$
– Math_QED
Dec 1 '18 at 16:19
1
$begingroup$
True, but you can apply it to the real and imaginary parts of your measure.
$endgroup$
– Martin Argerami
Dec 1 '18 at 17:20
$begingroup$
Thanks for your answer. Yosida seems to define complex measure as a set function $varphi$ that is $sigma$-additive and such that $|varphi(B)| neq infty$ for all $B$, so I'm not entirely sure if you completely answered the question.
$endgroup$
– Math_QED
Dec 1 '18 at 16:02
$begingroup$
Thanks for your answer. Yosida seems to define complex measure as a set function $varphi$ that is $sigma$-additive and such that $|varphi(B)| neq infty$ for all $B$, so I'm not entirely sure if you completely answered the question.
$endgroup$
– Math_QED
Dec 1 '18 at 16:02
$begingroup$
Moreover, the Lebesgue measure has total variation $infty$ so the theorem would exclude it anyway.
$endgroup$
– Math_QED
Dec 1 '18 at 16:04
$begingroup$
Moreover, the Lebesgue measure has total variation $infty$ so the theorem would exclude it anyway.
$endgroup$
– Math_QED
Dec 1 '18 at 16:04
$begingroup$
That's my point, the theorem excludes Lebesgue measure because it doesn't have finite total variation. Rudin would exclude it by definition. Now, if Yosida defines complex measure the same as Rudin, there is no need to require the hypothesis.
$endgroup$
– Martin Argerami
Dec 1 '18 at 16:14
$begingroup$
That's my point, the theorem excludes Lebesgue measure because it doesn't have finite total variation. Rudin would exclude it by definition. Now, if Yosida defines complex measure the same as Rudin, there is no need to require the hypothesis.
$endgroup$
– Martin Argerami
Dec 1 '18 at 16:14
$begingroup$
I don't know if you have a copy of Yosida at hand or even own the book, but I think he defines complex measures on p35. He uses an alternative definition, which I believe is equivalent with Rudin's. This equivalence is an exercise in Folland's real analysis. It sure is confusing haha.
$endgroup$
– Math_QED
Dec 1 '18 at 16:19
$begingroup$
I don't know if you have a copy of Yosida at hand or even own the book, but I think he defines complex measures on p35. He uses an alternative definition, which I believe is equivalent with Rudin's. This equivalence is an exercise in Folland's real analysis. It sure is confusing haha.
$endgroup$
– Math_QED
Dec 1 '18 at 16:19
1
1
$begingroup$
True, but you can apply it to the real and imaginary parts of your measure.
$endgroup$
– Martin Argerami
Dec 1 '18 at 17:20
$begingroup$
True, but you can apply it to the real and imaginary parts of your measure.
$endgroup$
– Martin Argerami
Dec 1 '18 at 17:20
|
show 5 more comments
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