2

I'm teaching a real analysis course and I'm trying to find a video or picture or app that illustrates the Lebesgue integral of a nonnegative function as the "supremum of integrals of smaller simple functions."

I know Lebesgue Integral - graphical concept has some stuff like above. I really do like https://i.stack.imgur.com/loFC7.gif but graphically I think students would be confused by the graphical restriction from $\alpha$ to $\beta$ (and even I myself don't quite get it). Also https://i.stack.imgur.com/P2jqY.gif isn't bad, but I think a parabola example or something NOT monotone illustrates the concept better. Lastly I found https://iwant2study.org/lookangejss/math/Calculus/ejss_model_e_Lebesgue_integral/e_Lebesgue_integral_Simulation.xhtml but this confuses me a bit too...

Finally, I can find plenty of "horizontal rectangle" pictures and videos but this obviously is an example of the equivalent "integrate the measure of level sets..." definition. I'm personally quite fond of estimating integrals using this (and use this in my research often, particularly the $L^p$ norm version), but conceptually this is much different than the definition of the Lebesgue integral in the first paragraph and I think this should be taught as a beautiful theorem proved using Fubini's theorem, rather than "the defintion".

Maybe I just need to write my own app? :(

NOTE: After writing this, I did find https://i.stack.imgur.com/ht5Fr.png on On the horizontal integration of the Lebesgue integral which looks really nice, but I'd REALLY like to modify the "y axis partition size"

Joshua Isralowitz
  • 95

1 Answers1

2

I'd REALLY like to modify the "y axis partition size"

Well, try these dynamic pictures:

  1. https://www.geogebra.org/m/emzhfjpf. Here, you can choose the number of equal partitions of the range by dragging $n$.

  2. https://www.geogebra.org/m/urhcb98v. Here, you can change the size of each partition by dragging the points on the $y$-axis.

In both cases:

  • You can change the shape of the graph by checking the box and dragging the points on the curve.
  • The simple function was defined by $\alpha_i(x)=\min\{f(a_i),f(b_i),0\}$ for all $x\in (a_i,b_i)$.

If you notice any unexpected behavior, let me know and I'll try to fix it (when I have time). Of course, you can also download the file and make any modifications you want.

Pedro
  • 18,817
  • 7
  • 65
  • 127