https://machinelearning.subwiki.org/w/index.php?action=history&feed=atom&title=User%3AIssaRice%2FLebesgue_theoryUser:IssaRice/Lebesgue theory - Revision history2026-06-23T23:44:10ZRevision history for this page on the wikiMediaWiki 1.41.2https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Lebesgue_theory&diff=3572&oldid=prevIssaRice at 20:37, 8 April 20232023-04-08T20:37:57Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* what if we used simple functions to define the riemann integral, or piecewise constant functions to define the lebesgue integral?</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* what if we used simple functions to define the riemann integral, or piecewise constant functions to define the lebesgue integral?</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* i think a lot of definitions of the lebesgue integral "cheat" by using the lebesgue measure (it puts all the difficulty of measurement into the lebesgue measure part). the riemann integral doesn't use the jordan measure in most definitions; you just add up the rectangles yourself. can we just get the lebesgue integral by using a countable number of rectangles? or is there more to lebesgue than that? can we make use of measurability without using the measure itself, to define the lebesgue integral?</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* i think a lot of definitions of the lebesgue integral "cheat" by using the lebesgue measure (it puts all the difficulty of measurement into the lebesgue measure part). the riemann integral doesn't use the jordan measure in most definitions; you just add up the rectangles yourself. can we just get the lebesgue integral by using a countable number of rectangles? or is there more to lebesgue than that? can we make use of measurability without using the measure itself, to define the lebesgue integral?</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* the horizontal slab idea is coming from [https://en.wikipedia.org/wiki/Lebesgue_integration#Via_improper_Riemann_integral this definition]. the weird thing though is that almost no one defines the lebesgue integral that way??? so like, the image you present has nothing to do with the technical definition you give. also, the horizontal slab thing is just using the lebesgue measure! all the work is being done by the lebesgue measure! if you substituted the jordan measure instead then i think you would just get back the riemann integral again.</ins></div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>pugh's book has more connections between riemann and lebesgue</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>pugh's book has more connections between riemann and lebesgue</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Lebesgue_theory&diff=3475&oldid=prevIssaRice at 19:38, 3 October 20212021-10-03T19:38:44Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* if all lebesgue did was to replace "finite" with "countable" in the definition of jordan measure, why was lebesgue's theory considered so important/revolutionary? was it the fact that lebesgue also did all the legwork to prove that his measure/integral had all these nice properties?</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* if all lebesgue did was to replace "finite" with "countable" in the definition of jordan measure, why was lebesgue's theory considered so important/revolutionary? was it the fact that lebesgue also did all the legwork to prove that his measure/integral had all these nice properties?</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* what if we used simple functions to define the riemann integral, or piecewise constant functions to define the lebesgue integral?</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* what if we used simple functions to define the riemann integral, or piecewise constant functions to define the lebesgue integral?</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* i think a lot of definitions of the lebesgue integral "cheat" by using the lebesgue measure (it puts all the difficulty of measurement into the lebesgue measure part). the riemann integral doesn't use the jordan measure in most definitions; you just add up the rectangles yourself. can we just get the lebesgue integral by using a countable number of rectangles? or is there more to lebesgue than that? can we make use of measurability without using the measure itself, to define the lebesgue integral?</ins></div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>pugh's book has more connections between riemann and lebesgue</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>pugh's book has more connections between riemann and lebesgue</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Lebesgue_theory&diff=3464&oldid=prevIssaRice at 19:28, 29 September 20212021-09-29T19:28:17Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* more general way of generating questions: there's some stuff we talk about a lot in "riemann/jordan land" (e.g. upper and lower sums and defining integrability when the two are equal), and some stuff we talk about a lot in "lebesgue land" (e.g. caratheodory criterion). for each thing we talk about in one of the lands, what is the analogue of it in the other land?</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* more general way of generating questions: there's some stuff we talk about a lot in "riemann/jordan land" (e.g. upper and lower sums and defining integrability when the two are equal), and some stuff we talk about a lot in "lebesgue land" (e.g. caratheodory criterion). for each thing we talk about in one of the lands, what is the analogue of it in the other land?</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* if the lebesgue integral was just the riemann integral but dividing along the y-axis instead of the x-axis, then we should just be able to get the lebesgue integral by integrating f^-1 or something?</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* if the lebesgue integral was just the riemann integral but dividing along the y-axis instead of the x-axis, then we should just be able to get the lebesgue integral by integrating f^-1 or something?</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* the horizontal stacking picture people like to draw for the lebesgue integral seems deceptive? like, if you actually look at how a simple function is integrated, you kind of draw the horizontal "window" but only to find where the corresponding points are on the x-axis? and then you multiply the y value with the width along the x-axis, so the actual area for that bit is still a vertical rectangle.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* the horizontal stacking picture people like to draw for the lebesgue integral seems deceptive? like, if you actually look at how a simple function is integrated, you kind of draw the horizontal "window" but only to find where the corresponding points are on the x-axis? and then you multiply the y value with the width along the x-axis, so the actual area for that bit is still a vertical rectangle. <ins style="font-weight: bold; text-decoration: none;">thank god, someone else [https://www.youtube.com/watch?v=LDNDTOVnKJk&lc=UgzgAWpd40oYJpsTfbx4AaABAg noticed this]: "I think you haven't understood the Lebesgue integral at all. The animation with horizontal rectangles is quite flawed. That's not how it works!! Rectangles are vertical, what happens is that we split the range of the image into different sections. In each of them, we take an arbitrary point, and multiply it by the inverse function of the section. That may be one or more VERTICAL rectangles."</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* my [https://www.youtube.com/watch?v=LDNDTOVnKJk&lc=UgyBsn4N7-nI9GjKbU54AaABAg comment]: "If the only difference between the Riemann and Lebesgue integral was dividing up along the x-axis vs y-axis, then the Jordan measure (which uses boxes and doesn't care about the x or y axis) should be able to find the area under the graph of any Lebesgue-integrable function, right? The fact that this is not possible I think means there is some other deeper difference between the two integrals."</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* my [https://www.youtube.com/watch?v=LDNDTOVnKJk&lc=UgyBsn4N7-nI9GjKbU54AaABAg comment]: "If the only difference between the Riemann and Lebesgue integral was dividing up along the x-axis vs y-axis, then the Jordan measure (which uses boxes and doesn't care about the x or y axis) should be able to find the area under the graph of any Lebesgue-integrable function, right? The fact that this is not possible I think means there is some other deeper difference between the two integrals."</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* trying to measure the set of irrational numbers in [0,1]: i think this should have length 1, but even though it's a bounded set, the sup of the inner measure seems to be 0? actually it does seem to be possible to construct a positive measure subset that only contains irrationals [https://math.stackexchange.com/a/3932309/35525], which is pretty unintuitive. hmm, but this is still different from inner measure i think, because we aren't using a countable number of boxes inside the irrationals.. instead we're starting by surrounding the rationals and then substracting out those sets.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* trying to measure the set of irrational numbers in [0,1]: i think this should have length 1, but even though it's a bounded set, the sup of the inner measure seems to be 0? actually it does seem to be possible to construct a positive measure subset that only contains irrationals [https://math.stackexchange.com/a/3932309/35525], which is pretty unintuitive. hmm, but this is still different from inner measure i think, because we aren't using a countable number of boxes inside the irrationals.. instead we're starting by surrounding the rationals and then substracting out those sets.</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Lebesgue_theory&diff=3463&oldid=prevIssaRice at 19:23, 29 September 20212021-09-29T19:23:26Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* ok here's another reason i think the y-axis thing is bullshit: we could define a "jordan integral" by taking the definition of the lebesgue integral via simple functions but replacing the measure with the jordan measure. i think the resulting integral would be equivalent to the riemann integral.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* ok here's another reason i think the y-axis thing is bullshit: we could define a "jordan integral" by taking the definition of the lebesgue integral via simple functions but replacing the measure with the jordan measure. i think the resulting integral would be equivalent to the riemann integral.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* if all lebesgue did was to replace "finite" with "countable" in the definition of jordan measure, why was lebesgue's theory considered so important/revolutionary? was it the fact that lebesgue also did all the legwork to prove that his measure/integral had all these nice properties?</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* if all lebesgue did was to replace "finite" with "countable" in the definition of jordan measure, why was lebesgue's theory considered so important/revolutionary? was it the fact that lebesgue also did all the legwork to prove that his measure/integral had all these nice properties?</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* what if we used simple functions to define the riemann integral, or piecewise constant functions to define the lebesgue integral?</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>pugh's book has more connections between riemann and lebesgue</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>pugh's book has more connections between riemann and lebesgue</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Lebesgue_theory&diff=3441&oldid=prevIssaRice at 17:21, 16 September 20212021-09-16T17:21:12Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** is there an analogue of the caratheodory theorem for jordan measure, to allow us to extend jordan measurability to unbounded sets? or is this not an interesting question to ask since if we try to measure such sets, the answer will always be infinity even for "thin" sets like Q?</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** is there an analogue of the caratheodory theorem for jordan measure, to allow us to extend jordan measurability to unbounded sets? or is this not an interesting question to ask since if we try to measure such sets, the answer will always be infinity even for "thin" sets like Q?</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* i've always found the coin-counting analogy between riemann and lebesgue measure confusing. why should one method be better than the other, when they both produce the same answer? if i was actually trying to quickly estimate how much i had in coins, i would just gather up all the highest denominations and count those, then add a "fuzz factor" to account for some error. i wouldn't even bother counting the pennies unless it seemed like there was a huge number of them.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* i've always found the coin-counting analogy between riemann and lebesgue measure confusing. why should one method be better than the other, when they both produce the same answer? if i was actually trying to quickly estimate how much i had in coins, i would just gather up all the highest denominations and count those, then add a "fuzz factor" to account for some error. i wouldn't even bother counting the pennies unless it seemed like there was a huge number of them.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* the riemann/jordan vs lebesgue difference has been described as finite vs countable, x-axis partitioning vs y-axis partitioning, topological boundary measure = 0 vs measure-theoretic boundary measure = 0, and maybe one other thing i am forgetting. but what is the essence of the difference here? why do all these separate distinctions turn out to be "the same thing"?</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* the riemann/jordan vs lebesgue difference has been described as finite vs countable, x-axis partitioning vs y-axis partitioning, topological boundary measure = 0 vs measure-theoretic boundary measure = 0, <ins style="font-weight: bold; text-decoration: none;">piecewise constant function vs simple function </ins>and maybe one other thing i am forgetting. but what is the essence of the difference here? why do all these separate distinctions turn out to be "the same thing"?</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** why should partitioning the y-axis lead to being able to integrate more functions compared to partitioning the x-axis? the finite vs countable distinction makes sense, but the y vs x thing makes no sense to me.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>** why should partitioning the y-axis lead to being able to integrate more functions compared to partitioning the x-axis? the finite vs countable distinction makes sense, but the y vs x thing makes no sense to me.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Apostol's analysis lists two non-equivalent definitions of riemann integrability i think. which one does the jordan-undergraph riemann integral pick out and why?</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Apostol's analysis lists two non-equivalent definitions of riemann integrability i think. which one does the jordan-undergraph riemann integral pick out and why?</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Lebesgue_theory&diff=3440&oldid=prevIssaRice at 07:29, 16 September 20212021-09-16T07:29:11Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 07:29, 16 September 2021</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* trying to measure the set of irrational numbers in [0,1]: i think this should have length 1, but even though it's a bounded set, the sup of the inner measure seems to be 0? actually it does seem to be possible to construct a positive measure subset that only contains irrationals [https://math.stackexchange.com/a/3932309/35525], which is pretty unintuitive. hmm, but this is still different from inner measure i think, because we aren't using a countable number of boxes inside the irrationals.. instead we're starting by surrounding the rationals and then substracting out those sets.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* trying to measure the set of irrational numbers in [0,1]: i think this should have length 1, but even though it's a bounded set, the sup of the inner measure seems to be 0? actually it does seem to be possible to construct a positive measure subset that only contains irrationals [https://math.stackexchange.com/a/3932309/35525], which is pretty unintuitive. hmm, but this is still different from inner measure i think, because we aren't using a countable number of boxes inside the irrationals.. instead we're starting by surrounding the rationals and then substracting out those sets.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* ok here's another reason i think the y-axis thing is bullshit: we could define a "jordan integral" by taking the definition of the lebesgue integral via simple functions but replacing the measure with the jordan measure. i think the resulting integral would be equivalent to the riemann integral.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* ok here's another reason i think the y-axis thing is bullshit: we could define a "jordan integral" by taking the definition of the lebesgue integral via simple functions but replacing the measure with the jordan measure. i think the resulting integral would be equivalent to the riemann integral.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* if all lebesgue did was to replace "finite" with "countable" in the definition of jordan measure, why was lebesgue's theory considered so important/revolutionary? was it the fact that lebesgue also did all the legwork to prove that his measure/integral had all these nice properties?</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>pugh's book has more connections between riemann and lebesgue</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>pugh's book has more connections between riemann and lebesgue</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Lebesgue_theory&diff=3439&oldid=prevIssaRice at 02:11, 16 September 20212021-09-16T02:11:59Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 02:11, 16 September 2021</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l20">Line 20:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* the horizontal stacking picture people like to draw for the lebesgue integral seems deceptive? like, if you actually look at how a simple function is integrated, you kind of draw the horizontal "window" but only to find where the corresponding points are on the x-axis? and then you multiply the y value with the width along the x-axis, so the actual area for that bit is still a vertical rectangle.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* the horizontal stacking picture people like to draw for the lebesgue integral seems deceptive? like, if you actually look at how a simple function is integrated, you kind of draw the horizontal "window" but only to find where the corresponding points are on the x-axis? and then you multiply the y value with the width along the x-axis, so the actual area for that bit is still a vertical rectangle.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* my [https://www.youtube.com/watch?v=LDNDTOVnKJk&lc=UgyBsn4N7-nI9GjKbU54AaABAg comment]: "If the only difference between the Riemann and Lebesgue integral was dividing up along the x-axis vs y-axis, then the Jordan measure (which uses boxes and doesn't care about the x or y axis) should be able to find the area under the graph of any Lebesgue-integrable function, right? The fact that this is not possible I think means there is some other deeper difference between the two integrals."</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* my [https://www.youtube.com/watch?v=LDNDTOVnKJk&lc=UgyBsn4N7-nI9GjKbU54AaABAg comment]: "If the only difference between the Riemann and Lebesgue integral was dividing up along the x-axis vs y-axis, then the Jordan measure (which uses boxes and doesn't care about the x or y axis) should be able to find the area under the graph of any Lebesgue-integrable function, right? The fact that this is not possible I think means there is some other deeper difference between the two integrals."</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* trying to measure the set of irrational numbers in [0,1]: i think this should have length 1, but even though it's a bounded set, the sup of the inner measure seems to be 0? actually it does seem to be possible to construct a positive measure subset that only contains irrationals [https://math.stackexchange.com/a/3932309/35525], which is pretty unintuitive.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* trying to measure the set of irrational numbers in [0,1]: i think this should have length 1, but even though it's a bounded set, the sup of the inner measure seems to be 0? actually it does seem to be possible to construct a positive measure subset that only contains irrationals [https://math.stackexchange.com/a/3932309/35525], which is pretty unintuitive<ins style="font-weight: bold; text-decoration: none;">. hmm, but this is still different from inner measure i think, because we aren't using a countable number of boxes inside the irrationals.. instead we're starting by surrounding the rationals and then substracting out those sets</ins>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* ok here's another reason i think the y-axis thing is bullshit: we could define a "jordan integral" by taking the definition of the lebesgue integral via simple functions but replacing the measure with the jordan measure. i think the resulting integral would be equivalent to the riemann integral.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* ok here's another reason i think the y-axis thing is bullshit: we could define a "jordan integral" by taking the definition of the lebesgue integral via simple functions but replacing the measure with the jordan measure. i think the resulting integral would be equivalent to the riemann integral.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Lebesgue_theory&diff=3438&oldid=prevIssaRice at 02:07, 16 September 20212021-09-16T02:07:22Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 02:07, 16 September 2021</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* the horizontal stacking picture people like to draw for the lebesgue integral seems deceptive? like, if you actually look at how a simple function is integrated, you kind of draw the horizontal "window" but only to find where the corresponding points are on the x-axis? and then you multiply the y value with the width along the x-axis, so the actual area for that bit is still a vertical rectangle.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* the horizontal stacking picture people like to draw for the lebesgue integral seems deceptive? like, if you actually look at how a simple function is integrated, you kind of draw the horizontal "window" but only to find where the corresponding points are on the x-axis? and then you multiply the y value with the width along the x-axis, so the actual area for that bit is still a vertical rectangle.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* my [https://www.youtube.com/watch?v=LDNDTOVnKJk&lc=UgyBsn4N7-nI9GjKbU54AaABAg comment]: "If the only difference between the Riemann and Lebesgue integral was dividing up along the x-axis vs y-axis, then the Jordan measure (which uses boxes and doesn't care about the x or y axis) should be able to find the area under the graph of any Lebesgue-integrable function, right? The fact that this is not possible I think means there is some other deeper difference between the two integrals."</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* my [https://www.youtube.com/watch?v=LDNDTOVnKJk&lc=UgyBsn4N7-nI9GjKbU54AaABAg comment]: "If the only difference between the Riemann and Lebesgue integral was dividing up along the x-axis vs y-axis, then the Jordan measure (which uses boxes and doesn't care about the x or y axis) should be able to find the area under the graph of any Lebesgue-integrable function, right? The fact that this is not possible I think means there is some other deeper difference between the two integrals."</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* trying to measure the set of irrational numbers in [0,1]: i think this should have length 1, but even though it's a bounded set, the sup of the inner measure seems to be 0?</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* trying to measure the set of irrational numbers in [0,1]: i think this should have length 1, but even though it's a bounded set, the sup of the inner measure seems to be 0? <ins style="font-weight: bold; text-decoration: none;">actually it does seem to be possible to construct a positive measure subset that only contains irrationals [https://math.stackexchange.com/a/3932309/35525], which is pretty unintuitive.</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* ok here's another reason i think the y-axis thing is bullshit: we could define a "jordan integral" by taking the definition of the lebesgue integral via simple functions but replacing the measure with the jordan measure. i think the resulting integral would be equivalent to the riemann integral.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* ok here's another reason i think the y-axis thing is bullshit: we could define a "jordan integral" by taking the definition of the lebesgue integral via simple functions but replacing the measure with the jordan measure. i think the resulting integral would be equivalent to the riemann integral.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Lebesgue_theory&diff=3437&oldid=prevIssaRice at 23:27, 15 September 20212021-09-15T23:27:17Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* my [https://www.youtube.com/watch?v=LDNDTOVnKJk&lc=UgyBsn4N7-nI9GjKbU54AaABAg comment]: "If the only difference between the Riemann and Lebesgue integral was dividing up along the x-axis vs y-axis, then the Jordan measure (which uses boxes and doesn't care about the x or y axis) should be able to find the area under the graph of any Lebesgue-integrable function, right? The fact that this is not possible I think means there is some other deeper difference between the two integrals."</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* my [https://www.youtube.com/watch?v=LDNDTOVnKJk&lc=UgyBsn4N7-nI9GjKbU54AaABAg comment]: "If the only difference between the Riemann and Lebesgue integral was dividing up along the x-axis vs y-axis, then the Jordan measure (which uses boxes and doesn't care about the x or y axis) should be able to find the area under the graph of any Lebesgue-integrable function, right? The fact that this is not possible I think means there is some other deeper difference between the two integrals."</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* trying to measure the set of irrational numbers in [0,1]: i think this should have length 1, but even though it's a bounded set, the sup of the inner measure seems to be 0?</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* trying to measure the set of irrational numbers in [0,1]: i think this should have length 1, but even though it's a bounded set, the sup of the inner measure seems to be 0?</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* ok here's another reason i think the y-axis thing is bullshit: we could define a "jordan integral" by taking the lebesgue integral but replacing the measure with the jordan measure. i think the resulting integral would be equivalent to the riemann integral.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* ok here's another reason i think the y-axis thing is bullshit: we could define a "jordan integral" by taking <ins style="font-weight: bold; text-decoration: none;">the definition of </ins>the lebesgue integral <ins style="font-weight: bold; text-decoration: none;">via simple functions </ins>but replacing the measure with the jordan measure. i think the resulting integral would be equivalent to the riemann integral.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>pugh's book has more connections between riemann and lebesgue</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>pugh's book has more connections between riemann and lebesgue</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Lebesgue_theory&diff=3436&oldid=prevIssaRice at 23:26, 15 September 20212021-09-15T23:26:41Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:26, 15 September 2021</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l21">Line 21:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* my [https://www.youtube.com/watch?v=LDNDTOVnKJk&lc=UgyBsn4N7-nI9GjKbU54AaABAg comment]: "If the only difference between the Riemann and Lebesgue integral was dividing up along the x-axis vs y-axis, then the Jordan measure (which uses boxes and doesn't care about the x or y axis) should be able to find the area under the graph of any Lebesgue-integrable function, right? The fact that this is not possible I think means there is some other deeper difference between the two integrals."</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* my [https://www.youtube.com/watch?v=LDNDTOVnKJk&lc=UgyBsn4N7-nI9GjKbU54AaABAg comment]: "If the only difference between the Riemann and Lebesgue integral was dividing up along the x-axis vs y-axis, then the Jordan measure (which uses boxes and doesn't care about the x or y axis) should be able to find the area under the graph of any Lebesgue-integrable function, right? The fact that this is not possible I think means there is some other deeper difference between the two integrals."</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* trying to measure the set of irrational numbers in [0,1]: i think this should have length 1, but even though it's a bounded set, the sup of the inner measure seems to be 0?</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* trying to measure the set of irrational numbers in [0,1]: i think this should have length 1, but even though it's a bounded set, the sup of the inner measure seems to be 0?</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* ok here's another reason i think the y-axis thing is bullshit: we could define a "jordan integral" by taking the lebesgue integral but replacing the measure with the jordan measure. i think the resulting integral would be equivalent to the riemann integral.</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>pugh's book has more connections between riemann and lebesgue</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>pugh's book has more connections between riemann and lebesgue</div></td></tr>
</table>IssaRice