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I am trying to learn about Lebesgue integration and have some initial questions:

  • Why is Lebesgue integration used instead of Riemann integration? What are its benefits/disadvantages?
  • What does it mean for a set to have Lebesgue measure zero?
Enigma123
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    The second question can be looked up in millions of places online including wikipedia. – Jonas Meyer Oct 07 '17 at 17:36
  • There are no disadvantages. ;o) With the Lebesgue integral you can integrate much more functions than with the Riemannian. Also, you have many nice convergence theorems that you don't have for the Riemann integral. – Friedrich Philipp Oct 07 '17 at 17:36
  • The first can be researched in part on this website; putting keywords into the search box led to: https://math.stackexchange.com/questions/2218114/theoretical-advantages-of-lebesgue-integration, https://math.stackexchange.com/questions/1318557/benefit-from-measure-theory, https://math.stackexchange.com/q/480049/. Maybe you have a question not covered there and could elaborate what that is? – Jonas Meyer Oct 07 '17 at 17:41
  • Thank you both, @JonasMeyer I am looking more for an intuitive explanation of what a set with Lebesgue measure zero means rather than a formal definition of what Lebesgue measure zero is. I am struggling to visually understand what the measure actually is. – Enigma123 Oct 07 '17 at 18:03
  • Lebesgue proved that a bounded function is Riemann integrable on $[a,b]$ iff the function is continuous almost everywhere, a definition which led to the notion of measure zero and, generally, to the notion of the size of a subset of $\mathbb{R}$. Almost everywhere is defined by saying that, for every $\epsilon > 0$, the set is covered by finite or countable collection (depending on $\epsilon$) of open intervals whose lengths sum to something less than $\epsilon$. That covering led to the idea of measuring the size of a set of real numbers; Lebesgue used this to define his integral. – Disintegrating By Parts Oct 08 '17 at 11:22

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Advantages of the Riemann integral: quicker to define. Continuous functions on a finite closed interval have a R integral. It can be taught to high school students. In fact, it is easy to define the integral of a continuous function on $[a,b]$ with values in a Banach space. Try that with the Lebesgue integral in a quick manner.

I think the Lebesgue integral needs either measure theory predefined, or a bit of topology ( semi-continuous functions, etc). Not suitable for high school.

orangeskid
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  • The Lebesgue integral is the extension of the Riemann integral from the space of continuous functions to its completion. That’s a pretty elemntary definition of Lebesgue integral requiring no measure theory or tpology. – Ittay Weiss Oct 07 '17 at 18:29
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    @Ittay Weiss: yes, indeed, and if I remember correctly it takes some effort to see an element of the completion as a function ( defined a.e.). And how do you see right away that, say, the pointwise limit of a sequence of continuous functions from $[0,1]$ to $[0,1]$ is in that completion? I don't think pure functional analysis does it right away. Post factum, sure. Unless my measure theory is really rusty... – orangeskid Oct 07 '17 at 18:39
  • With this approach you don’t need to think of an element in the completion as an equivalence class of functions. As for the pointwise limit, it’s a Cauchy sequence. – Ittay Weiss Oct 07 '17 at 18:52