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Intuitively, why does a unique value for the Henstock-Kurzweil (gauge) integral always exist?
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A(n improperly) HK-integrable function $f$ is Lebesgue integrable on a (not necessarily bounded) measurable subset of $mathbbR$ if and only if $|f|$ is also (improperly) HK-integrable on that subset.
Found the above claim at the following sources: (1)(2)(3)(4).
This means that if $f$ is a(n improperly) HK-integrable function whichi is not Lebesgue integrable, then $|f|$ is not (improperly) HK-integrable. Call such functions "conditionally" HK-integrable.
Question: Intuitively speaking$^*$, why are the integrals of conditionally HK-integrable functions well defined? Why don't we have to define something like the Cauchy principal value?
$^*$ I.e. feel free to skip over as many details as you want. I don't understand the HK integral well.
As you may have guessed from my decision to call such functions "conditionally HK-integrable", my intuition is that they should be analogous to conditionally convergent series. Therefore, intuitively I would expect a problem resulting from $infty - infty$ not being possible to define well, leading to an analog of the Riemann rearrangement theorem being applicable, and thus to some convention like the Cauchy principal value being necessary to ever assign any value to the integral at all.
If $|f| = f^+ + f^-$ is not HK-integrable, but $f = f^+ - f^-$ is HK-integrable (i.e. $f$ is "conditionally" HK-integrable), then this seems to require that both $HKint f^+ = + infty = HK int f^-$. Not understanding the definition of HK integral well, however, I am not sure if this argument works.
Motivation: If the argument does work, then since the only HK-integrable functions which aren't Lebesgue integrable are conditionally HK-integrable, then we would have another answer to this popular question (see also this question) (and also this question).
The answer to the following question is related to this question: Can someone give an example of a function that is not Henstock-Kurzweil/gauge integrable?
Seemingly also related: Can conditionally convergent series be interpreted as a "generalized Henstock-Kurzweil integral"?
real-analysis integration improper-integrals intuition
asked Jul 15 at 1:08
Chill2Macht
12.1k91659
add a comment |Â
up vote
1
down vote
favorite
A(n improperly) HK-integrable function $f$ is Lebesgue integrable on a (not necessarily bounded) measurable subset of $mathbbR$ if and only if $|f|$ is also (improperly) HK-integrable on that subset.
Found the above claim at the following sources: (1)(2)(3)(4).
This means that if $f$ is a(n improperly) HK-integrable function whichi is not Lebesgue integrable, then $|f|$ is not (improperly) HK-integrable. Call such functions "conditionally" HK-integrable.
Question: Intuitively speaking$^*$, why are the integrals of conditionally HK-integrable functions well defined? Why don't we have to define something like the Cauchy principal value?
$^*$ I.e. feel free to skip over as many details as you want. I don't understand the HK integral well.
As you may have guessed from my decision to call such functions "conditionally HK-integrable", my intuition is that they should be analogous to conditionally convergent series. Therefore, intuitively I would expect a problem resulting from $infty - infty$ not being possible to define well, leading to an analog of the Riemann rearrangement theorem being applicable, and thus to some convention like the Cauchy principal value being necessary to ever assign any value to the integral at all.
If $|f| = f^+ + f^-$ is not HK-integrable, but $f = f^+ - f^-$ is HK-integrable (i.e. $f$ is "conditionally" HK-integrable), then this seems to require that both $HKint f^+ = + infty = HK int f^-$. Not understanding the definition of HK integral well, however, I am not sure if this argument works.
Motivation: If the argument does work, then since the only HK-integrable functions which aren't Lebesgue integrable are conditionally HK-integrable, then we would have another answer to this popular question (see also this question) (and also this question).
The answer to the following question is related to this question: Can someone give an example of a function that is not Henstock-Kurzweil/gauge integrable?
Seemingly also related: Can conditionally convergent series be interpreted as a "generalized Henstock-Kurzweil integral"?
real-analysis integration improper-integrals intuition
asked Jul 15 at 1:08
Chill2Macht
12.1k91659
Did you try to see what happens in the gauge level when you try to prove, for example, that $sum_n (-1)^n/n$ is convergent in the language of Henstock integral?
– Hugocito
Jul 15 at 1:24
Reading T.Kunkle Rearrangements of Conditionally Integrable Functions 1994 can be useful.
– Tony Piccolo
Jul 19 at 16:27
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
A(n improperly) HK-integrable function $f$ is Lebesgue integrable on a (not necessarily bounded) measurable subset of $mathbbR$ if and only if $|f|$ is also (improperly) HK-integrable on that subset.
Found the above claim at the following sources: (1)(2)(3)(4).
This means that if $f$ is a(n improperly) HK-integrable function whichi is not Lebesgue integrable, then $|f|$ is not (improperly) HK-integrable. Call such functions "conditionally" HK-integrable.
Question: Intuitively speaking$^*$, why are the integrals of conditionally HK-integrable functions well defined? Why don't we have to define something like the Cauchy principal value?
$^*$ I.e. feel free to skip over as many details as you want. I don't understand the HK integral well.
As you may have guessed from my decision to call such functions "conditionally HK-integrable", my intuition is that they should be analogous to conditionally convergent series. Therefore, intuitively I would expect a problem resulting from $infty - infty$ not being possible to define well, leading to an analog of the Riemann rearrangement theorem being applicable, and thus to some convention like the Cauchy principal value being necessary to ever assign any value to the integral at all.
If $|f| = f^+ + f^-$ is not HK-integrable, but $f = f^+ - f^-$ is HK-integrable (i.e. $f$ is "conditionally" HK-integrable), then this seems to require that both $HKint f^+ = + infty = HK int f^-$. Not understanding the definition of HK integral well, however, I am not sure if this argument works.
Motivation: If the argument does work, then since the only HK-integrable functions which aren't Lebesgue integrable are conditionally HK-integrable, then we would have another answer to this popular question (see also this question) (and also this question).
The answer to the following question is related to this question: Can someone give an example of a function that is not Henstock-Kurzweil/gauge integrable?
Seemingly also related: Can conditionally convergent series be interpreted as a "generalized Henstock-Kurzweil integral"?
real-analysis integration improper-integrals intuition
asked Jul 15 at 1:08
Chill2Macht
12.1k91659
A(n improperly) HK-integrable function $f$ is Lebesgue integrable on a (not necessarily bounded) measurable subset of $mathbbR$ if and only if $|f|$ is also (improperly) HK-integrable on that subset.
Found the above claim at the following sources: (1)(2)(3)(4).
This means that if $f$ is a(n improperly) HK-integrable function whichi is not Lebesgue integrable, then $|f|$ is not (improperly) HK-integrable. Call such functions "conditionally" HK-integrable.
Question: Intuitively speaking$^*$, why are the integrals of conditionally HK-integrable functions well defined? Why don't we have to define something like the Cauchy principal value?
$^*$ I.e. feel free to skip over as many details as you want. I don't understand the HK integral well.
As you may have guessed from my decision to call such functions "conditionally HK-integrable", my intuition is that they should be analogous to conditionally convergent series. Therefore, intuitively I would expect a problem resulting from $infty - infty$ not being possible to define well, leading to an analog of the Riemann rearrangement theorem being applicable, and thus to some convention like the Cauchy principal value being necessary to ever assign any value to the integral at all.
If $|f| = f^+ + f^-$ is not HK-integrable, but $f = f^+ - f^-$ is HK-integrable (i.e. $f$ is "conditionally" HK-integrable), then this seems to require that both $HKint f^+ = + infty = HK int f^-$. Not understanding the definition of HK integral well, however, I am not sure if this argument works.
Motivation: If the argument does work, then since the only HK-integrable functions which aren't Lebesgue integrable are conditionally HK-integrable, then we would have another answer to this popular question (see also this question) (and also this question).
The answer to the following question is related to this question: Can someone give an example of a function that is not Henstock-Kurzweil/gauge integrable?
Seemingly also related: Can conditionally convergent series be interpreted as a "generalized Henstock-Kurzweil integral"?
real-analysis integration improper-integrals intuition
asked Jul 15 at 1:08
Chill2Macht
12.1k91659
asked Jul 15 at 1:08
Chill2Macht
12.1k91659
asked Jul 15 at 1:08
Chill2Macht
12.1k91659
asked Jul 15 at 1:08
Chill2Macht
12.1k91659
12.1k91659
Did you try to see what happens in the gauge level when you try to prove, for example, that $sum_n (-1)^n/n$ is convergent in the language of Henstock integral?
– Hugocito
Jul 15 at 1:24
Reading T.Kunkle Rearrangements of Conditionally Integrable Functions 1994 can be useful.
– Tony Piccolo
Jul 19 at 16:27
add a comment |Â
Did you try to see what happens in the gauge level when you try to prove, for example, that $sum_n (-1)^n/n$ is convergent in the language of Henstock integral?
– Hugocito
Jul 15 at 1:24
Reading T.Kunkle Rearrangements of Conditionally Integrable Functions 1994 can be useful.
– Tony Piccolo
Jul 19 at 16:27
Did you try to see what happens in the gauge level when you try to prove, for example, that $sum_n (-1)^n/n$ is convergent in the language of Henstock integral?
– Hugocito
Jul 15 at 1:24
Did you try to see what happens in the gauge level when you try to prove, for example, that $sum_n (-1)^n/n$ is convergent in the language of Henstock integral?
– Hugocito
Jul 15 at 1:24
Reading T.Kunkle Rearrangements of Conditionally Integrable Functions 1994 can be useful.
– Tony Piccolo
Jul 19 at 16:27
Reading T.Kunkle Rearrangements of Conditionally Integrable Functions 1994 can be useful.
– Tony Piccolo
Jul 19 at 16:27
add a comment |Â
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Did you try to see what happens in the gauge level when you try to prove, for example, that $sum_n (-1)^n/n$ is convergent in the language of Henstock integral?
– Hugocito
Jul 15 at 1:24
Reading T.Kunkle Rearrangements of Conditionally Integrable Functions 1994 can be useful.
– Tony Piccolo
Jul 19 at 16:27