What is the difference between Riemann-integrable and Lebesgue-integrable? Does it have anything to do with the absolute value of the integrand; something like
$\text{Lebesgue-integrable} \Leftrightarrow \int |f(x)| < \infty$?
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4$\begingroup$ No, it has to do with what we mean by the symbol $\int$. $\endgroup$– YoTengoUnLCDCommented Feb 5, 2016 at 16:56
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2$\begingroup$ Lesbegue integration allows us to integrate a much larger class of functions, such as $\chi_{\mathbb{Q}}$, the characteristic function of the rationals. It also has much nicer properties than the riemann integral. For instance, lesbegue integration commutes with limits. $\endgroup$– basketCommented Feb 5, 2016 at 16:57
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$\begingroup$ Could you explain further? How is the characteristic function of the rationals lebesgque-integrable and what is its value? I am a physicist so I know some math but by far not as much as I am supposed to. I am trying to really understand the differences between riemann and lebesgue integration but the Wikipedia article already is too hard for me :( $\endgroup$– AlphaOmegaCommented Feb 5, 2016 at 17:19
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2 Answers
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The main difference between integrability in the sense of Lebesgue and Riemann is the way we measure 'the area under the curve'.
The Riemann integral asks the question what's the 'height' of $f$ above a given part of the domain of the function. The Lebesgue integral on the other hand asks, for a given part of the range of $f$, what's the measure of the $x$'s which contribute to this 'height'.
The following is taken from the wikipedia page for Lebesgue integration, and most instructive (Riemann in blue on the top, Lebesgue in red on the bottom):
(Taken from https://en.wikipedia.org/wiki/Lebesgue_integration#/media/File:RandLintegrals.png, CC BY-SA 3.0)
Or, another way to explain this:
I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral.
This due to Reinhard Siegmund-Schultze, (2008), "Henri Lebesgue", in Timothy Gowers, June Barrow-Green, Imre Leader, Princeton Companion to Mathematics, Princeton University Press.
As a result of the different definitions, different classes of functions are integrable:
By definition, $f$ is Lebesgue-integrable iff $|f|$ is Lebesgue-integrable.
And, indeed, as basket noted, we can integrate a lot of functions in the Lebesgue sense which can't be integrated in the Riemann sense (e.g. the so-called Dirichlet function which is $1$ on the rational numbers and $0$ on the irrational ones). We even have the nice result that if $f$ is bounded and defined on a compact set and Riemann integrable, then it is Lebesgue integrable.
On the other hand, if the domain isn't bounded, then $f$ might be integrable in the improper Riemann sense, but not integrable in the Lebesgue sense (e.g. $f(x)=\frac{\sin(x)}{x}$ on $[1,\infty)$ - for this function, even $|f|$ is not integrable in the improper Riemann sense).
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1$\begingroup$ I think you mixed up the integrals in the picture. $\endgroup$ Commented Feb 5, 2016 at 17:51
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1$\begingroup$ @YoTengoUnLCD Indeed! Thanks, it's fixed now. $\endgroup$– RolandCommented Feb 5, 2016 at 18:25
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3$\begingroup$ Just wanted to point out that the "bills and coins" explanation was not due to Siegmund-Schultze (2008), who actually says that it is due to "Lebesgue himself, according to his colleague, Paul Montel". $\endgroup$– user46234Commented Oct 6, 2018 at 13:24
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3$\begingroup$ The “bills and coins” quote is great. The bottom image (the red image) does not reflect the quote and does not represent a Lebesgue integral - there are no horizontal rectangles in the conceptual view of the Lebesgue integral. Instead, the Lebesgue integral first locates all vertical rectangles corresponding to a given height, and then conceptually rearranges them to place them next to each other and count them as one single area; then does the same for all other heights. Just to save others wasted time trying to make the (erroneous) red image match the “coins and bills” quote. $\endgroup$ Commented Aug 5, 2023 at 11:12
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One can't really simplify things to the level that you apparently want. Simply the theory is somewhat complicated and there is no simple short answer to your question.
Since presumably you already know the definitions, let me comment on two aspects, which are essentially the most important, of course in MY opinion:
1) The simplest nontrivial comment is that Riemann-integration looks at the domain of the function, while Lebesgue-integration looks at the codomain, certainly at the expense of needing more than just intervals.
2) On the other hand, it is reasonable to argue that the most important property of Lebesgue-integration is that one can construct a complete space using (equivalence classes of) Lebesgue-integrable functions.
