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I was reading this beautiful discussion about the difference between Riemann and Lebesgue's integral .

Lebesgue integral basics

What is not clear to me is the following: if we are computing the area we are computing the portion of the graph of $y=f(x)$ that lies between the horizontal lines $y=y_i$ and $y=y_{i+1}$, why do we approximate it by using $ \sum_{i=0}^{n-1} \mu(E_i)y_i^*$? Should not it be $\sum_{i=1}^{n} \mu(E_i)(y_i- y_{i-1})$, where $y_i- y_{i-1}$ would be the "height" and the measure $\mu(E_i)$ , assuming the domain of $f$ is $R$, would be the "length"?

Edit:

For instance, suppose I am approximating the area under the curve in figure 2 below through the red rectangles. Why do not we use simply $\sum_{i=1}^{n} \mu(E_i)(y_i- y_{i-1})$? For instance, the area of the rectangle on top, should be simply $\mu(E_n)(y_n- y_{n-1})$. Where $\mu(E_n)$ is $(x:y_{n-1}<f(x)<y_n)$. What is wrong with my reasoning?

comparision of integrals

Alchemy
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  • The utility of the sideways rectangle method will be limited similarly to Riemann integration, possibly even more limited. How would use deal with a countable number of discontinuities, for example? Can you prove that the method you illustrate will work for any function with set of discontinuities being measure zero, for example, as in the Riemann-Lebesgue theorem? – jdods Oct 13 '19 at 06:29

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You have to realize that the $E_i$ is exactly the set of $x$-values where $y_i<f(x)\leq y_{i+1}$.

I find it more intuitive to think of $y^*_i$ as the height and $\mu(E_i)$ as the "length". It just happens that it might be the length of something that isn't and interval and therefor doesn't have a "natural" length. The length might need to be computed using more complicated techniques, hence the development of the Lebesgue "length" measure.

jdods
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  • Ok I understand, but why is the height $y_{i-1}<y^*i<y_i$ and not $y_i-y{i-1}$? We are computing the area of the graph of $y=f(x)$ that lies between the horizontal lines $y=y_i$ and $y=y_{i+1}$. This is what I do not get – Alchemy Oct 12 '19 at 12:28
  • All $x$ in domain set $E_i$ have height between $y_i$ and $y_{i+1}$. There are many different heights in there. We need to choose a single height to use over the entire set. That's the $y^_i$. This is where it's also a a bit different from Riemann, where the sample height comes from choosing an $x^$ and calculating $f(x^*)$. We don't worry about that, just choose a $y$ value between the desired range. – jdods Oct 12 '19 at 13:07
  • I have just edited to make it more clear what is my doubt. – Alchemy Oct 12 '19 at 14:44
  • Your picture with the red rectangles would be a valid way to approximate the area, but you need to define the set $E_i$ appropriately, and they will possibly be nested and not disjoint as is the case for both Riemann and Lebesgue integration. There is no benefit from the horizontal rectangle method though, probably. Whereas the benefit of Lebesgue measure is great. What you are illustrating is not Lebesgue integration, just to be clear. – jdods Oct 13 '19 at 06:05