The Integral That Changed Math Forever

Comments · 508

• AB

  @AbideByReason · 1 year ago · pinned

  To try everything Brilliant has to offer—free—for a full 30 days, visit <a href="https://brilliant.org/AbideByReason/">https://brilliant.org/AbideByReason/</a> . You’ll also get 20% off an annual premium subscription.

  52

• DA

  @DamnationGamesSM · 1 year ago (edited)

  I did math 411 (honors real analysis) in my fourth year and we learned Lebegue integration. First time in my life I got 0 on a midterm lmao

  91

• HE

  @henryginn7490 · 1 year ago

  Some of you may be thinking &quot;wait, if the length of a point is 0, and we use that to show the length of the rationals is 0, can&apos;t we do the same for the uncountables?&quot;. The resolution to this is because what &quot;length&quot; means is not covered in this video, probably because it has a pretty technical definition. The important thing to know is that the length of a union of disjoint sets is equal to the sum of the individual lengths for countably many sets, but not necessarily uncountably many. The rationals are countable so it applies, but the irrationals are not so we can&apos;t do infinity times 0 and get 0 (we do infinity times 0 and get 1, this is the strangeness of uncountability).

  818

• TU

  @tubebrocoli · 1 year ago

  i&quot;ve noticed that a lot of videos explaining the lebesgue integral hinge on saying that lebesgue &quot;just flipped the rectangles&quot;, but that&apos;s just incidental, the real progress came from lebesgue developing lebesgue measure, a rigorous way to measure sets. why the lebesgue integral is then built by summing the preimages times their values just follows naturally. the way it&apos;s presented it just looks arbitrary and nonsensical.

  436

• BA

  @basilemasson7683 · 1 year ago (edited)

  <a href="https://www.youtube.com/watch?v=Fb2ei6lD-d8&amp;t=494">8:14</a> theorem jumpscare

  72

• TY

  @Tyler-i2d · 1 year ago

  All of the math majors be like &apos;this video is missing context for definition of length and stuff like that&apos; but I feel like this video is made for someone like me, a physics major who is comfortable with calculus but lacks the proofy and real analysis background needed to jump into a formal discussion for measure theory. This vid is honestly really helping me break into a topic that I don&apos;t have alot of background context for, even if it is not presenting the ideas in the order that is most natural for a mathematician. good stuff!

  120

• JA

  @JayTemple · 1 year ago

  In Functional Analysis, I had learned some things earlier than others. When the professor gave an example of a function defined one way for the rationals and another for the irrationals, I asked, &quot;Couldn&apos;t you just say that since the rationals are a set of measure 0, the integral is whatever you get by taking the antiderivative of (the formula for the irrationals)?&quot; He said, &quot;You&apos;ve just described the difference between a Riemann integral and a Lebesgue integral.&quot; Until now I could never remember which one was the Lebesgue integral.

  9

• DO

  @douglasstrother6584 · 1 year ago

  My Calculus I &amp; II Professor (Tony Tromba, UC Santa Cruz, Fall 1981) dropped the Dirichlet Function on us at the end of a Friday lecture to give something to snack on during Happy Hour.

  75

• DI

  @dinhero21 · 1 year ago

  <a href="https://www.youtube.com/watch?v=Fb2ei6lD-d8&amp;t=347">5:47</a> aren&apos;t you missing the fact that irrationals and rationals are disjoint? you could use the same argument to prove that the length of [0,1] is 0, by showing that [0,1] is just the union of [0,1] with [0,1] so its length must be its length times 2, with the only number fitting that criterion being 0.

  4

• CL

  @ClementinesmWTF · 1 year ago

  To your point at the end about irrationals being the completion of the rationals: yes and no. In the sense that they are complements and the limit of rational sequences, absolutely. But in the sense that you also show using polynomials, no.<br><br>Polynomial solutions are a generalization of rationals called algebraic numbers that also includes irrational roots (eg sqrt(2)), complex values (eg quadratic integers), or can even be irrational numbers that are not expressible as closed forms (eg x|x^5-x-1=0).<br><br>Analogously to rationals having their limiting complement being the irrationals, the algebraic numbers have a really cool complement called the transcendental numbers (eg, pi, e). And similarly to the irrationals, the transcendental are also dense and have non-zero measure, which comes from the fact that algebraic numbers, much like their baby sibling the rationals, are only countably infinite.<br><br>This analogy can be further extended to show that non-computable numbers form almost the entire real number line and that what we interact with is only a very small portion of what is out there (precision be damned).

  13

• MI

  @mirastyle · 1 year ago

  I remember we had an entire semester class on measure &amp; integral going jus over this. It was intense 😂

  6

• KO

  @Koefferchenn · 1 year ago

  The argument that the lebesuque measure of the Rationals in [0,1] is 0 because each Individual point has measure 0 is not sufficient. ( <a href="https://www.youtube.com/watch?v=Fb2ei6lD-d8&amp;t=305">5:05</a> ) You could say the same about the irrational in [0,1] even though their measure is 1.

  71