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Hunter Spink
Hunter Spink
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For the entire question, the interval I am integrating over is $[0,1]$.

Background: In order to exhibit an isometry from $L^2[0,1]$ into $l^2$, I need to either assume absolute continuity over some interval for a whole family of functions due to the technicality of integrating a derivative using the Lebesgue integral, or I can use the Gauge integral.

I have very little experience with the Gauge integral, but if what I understand is correct, I can integrate the derivative of a function which exists at all but countably many points, and it behaves just like the Fundamental Theorem of Calculus says. However, I need to integrate $G_n'(x)$, where $G_n$ are $L^{1}[0,1]$ functions, which Rudin says in Thm. 7.14 of "Real and Complex Analysis" exists only almost everywhere.

Almost everywhere $\ne$ countable unfortunately, so my question is whether one of these two statements is correct:

i) Because I am working on a fixed bounded interval rather than $R^1$, I can assume that $G_n'(x)$ exists at all but countably many places (unlikely, and I believe shown false by the "distance to the cantor set" function)

ii) The Gauge integral of $G_n'(x)$ behaves as expected even over uncountable sets of measure 0.

If both of these are false, then my question is what is the least stringent condition I can impose on $G_n$ to ensure that the integration of the derivative works as expected. The actual functions $G_n$ are defined recursively, and even "absolute continuity" provides a condition too difficult to describe for the input function $G_0$.

Thank you very much MathOverflowers,

Hunter Spink

Edit: After a good night's sleep, I realize that both i) and ii) are rendered moot by functions like the cantor staircase (and it doesn't even make sense to ask about countably many discontinuities if working in $L^p$ spaces because functions are equivalent mod sets of measure 0). However, I am still interested in what conditions one needs to impose on a function to be able to recover itself just from its derivative (which I thought was absolute continuity everywhere except a set of measure 0 until I learned about the Gauge integral).

For the entire question, the interval I am integrating over is $[0,1]$.

Background: In order to exhibit an isometry from $L^2[0,1]$ into $l^2$, I need to either assume absolute continuity over some interval for a whole family of functions due to the technicality of integrating a derivative using the Lebesgue integral, or I can use the Gauge integral.

I have very little experience with the Gauge integral, but if what I understand is correct, I can integrate the derivative of a function which exists at all but countably many points, and it behaves just like the Fundamental Theorem of Calculus says. However, I need to integrate $G_n'(x)$, where $G_n$ are $L^{1}[0,1]$ functions, which Rudin says in Thm. 7.14 of "Real and Complex Analysis" exists only almost everywhere.

Almost everywhere $\ne$ countable unfortunately, so my question is whether one of these two statements is correct:

i) Because I am working on a fixed bounded interval rather than $R^1$, I can assume that $G_n'(x)$ exists at all but countably many places (unlikely, and I believe shown false by the "distance to the cantor set" function)

ii) The Gauge integral of $G_n'(x)$ behaves as expected even over uncountable sets of measure 0.

If both of these are false, then my question is what is the least stringent condition I can impose on $G_n$ to ensure that the integration of the derivative works as expected. The actual functions $G_n$ are defined recursively, and even "absolute continuity" provides a condition too difficult to describe for the input function $G_0$.

Thank you very much MathOverflowers,

Hunter Spink

For the entire question, the interval I am integrating over is $[0,1]$.

Background: In order to exhibit an isometry from $L^2[0,1]$ into $l^2$, I need to either assume absolute continuity over some interval for a whole family of functions due to the technicality of integrating a derivative using the Lebesgue integral, or I can use the Gauge integral.

I have very little experience with the Gauge integral, but if what I understand is correct, I can integrate the derivative of a function which exists at all but countably many points, and it behaves just like the Fundamental Theorem of Calculus says. However, I need to integrate $G_n'(x)$, where $G_n$ are $L^{1}[0,1]$ functions, which Rudin says in Thm. 7.14 of "Real and Complex Analysis" exists only almost everywhere.

Almost everywhere $\ne$ countable unfortunately, so my question is whether one of these two statements is correct:

i) Because I am working on a fixed bounded interval rather than $R^1$, I can assume that $G_n'(x)$ exists at all but countably many places (unlikely, and I believe shown false by the "distance to the cantor set" function)

ii) The Gauge integral of $G_n'(x)$ behaves as expected even over uncountable sets of measure 0.

If both of these are false, then my question is what is the least stringent condition I can impose on $G_n$ to ensure that the integration of the derivative works as expected. The actual functions $G_n$ are defined recursively, and even "absolute continuity" provides a condition too difficult to describe for the input function $G_0$.

Thank you very much MathOverflowers,

Hunter Spink

Edit: After a good night's sleep, I realize that both i) and ii) are rendered moot by functions like the cantor staircase (and it doesn't even make sense to ask about countably many discontinuities if working in $L^p$ spaces because functions are equivalent mod sets of measure 0). However, I am still interested in what conditions one needs to impose on a function to be able to recover itself just from its derivative (which I thought was absolute continuity everywhere except a set of measure 0 until I learned about the Gauge integral).

Source Link
Hunter Spink
Hunter Spink
  • 783
  • 1
  • 6
  • 10

Gauge integral of the derivative of a function except on a set of measure 0.

For the entire question, the interval I am integrating over is $[0,1]$.

Background: In order to exhibit an isometry from $L^2[0,1]$ into $l^2$, I need to either assume absolute continuity over some interval for a whole family of functions due to the technicality of integrating a derivative using the Lebesgue integral, or I can use the Gauge integral.

I have very little experience with the Gauge integral, but if what I understand is correct, I can integrate the derivative of a function which exists at all but countably many points, and it behaves just like the Fundamental Theorem of Calculus says. However, I need to integrate $G_n'(x)$, where $G_n$ are $L^{1}[0,1]$ functions, which Rudin says in Thm. 7.14 of "Real and Complex Analysis" exists only almost everywhere.

Almost everywhere $\ne$ countable unfortunately, so my question is whether one of these two statements is correct:

i) Because I am working on a fixed bounded interval rather than $R^1$, I can assume that $G_n'(x)$ exists at all but countably many places (unlikely, and I believe shown false by the "distance to the cantor set" function)

ii) The Gauge integral of $G_n'(x)$ behaves as expected even over uncountable sets of measure 0.

If both of these are false, then my question is what is the least stringent condition I can impose on $G_n$ to ensure that the integration of the derivative works as expected. The actual functions $G_n$ are defined recursively, and even "absolute continuity" provides a condition too difficult to describe for the input function $G_0$.

Thank you very much MathOverflowers,

Hunter Spink