Gauge integral of the derivative of a function except on a set of measure 0. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T17:04:43Z http://mathoverflow.net/feeds/question/75073 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75073/gauge-integral-of-the-derivative-of-a-function-except-on-a-set-of-measure-0 Gauge integral of the derivative of a function except on a set of measure 0. Rofler 2011-09-10T08:09:27Z 2011-09-10T17:38:45Z <p>For the entire question, the interval I am integrating over is \$[0,1]\$.</p> <p>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.</p> <p>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.</p> <p>Almost everywhere \$\ne\$ countable unfortunately, so my question is whether one of these two statements is correct:</p> <p>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)</p> <p>ii) The Gauge integral of \$G_n'(x)\$ behaves as expected even over uncountable sets of measure 0.</p> <p>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\$.</p> <p>Thank you very much MathOverflowers,</p> <p>Hunter Spink</p> <p>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).</p>