Teaching
Lebesgue vs Riemann Integration
Lebesgue Integration and Riemann Integration are rather different in flavour. As discussed in class, the fundamental difference is that Lebesgue integration uses a sequence of functions which approximate better and better the integral of the function of interest. In the interactive plot below we would like to integrate \(f(x) = \sin(x) \) on the interval \([0,\pi/2] \). The Lebesgue integral is constructed by consider the following series of approximating functions: $$ s_{n}(x) = \sum^{2^{n}}_{k=0} \sin\left(\frac{k}{2^{n+1}}\pi\right)\chi_{[\frac{k}{2^{n+1}}\pi,\frac{k+1}{2^{n+1}}\pi)}(x) $$ with \(n=1,2,\ldots\). The "Lebesgue Approximation" slider controls \(n\). The other two sliders, namely "Riemann Left Intervals"" and "Riemann Right Intervals", control the size of the partition of the Riemann Left and Right sums.Similarly, to approximate the integral \(\int^{1}_{0} \exp(x)dx \) we consider the following sequence of functions, $$ s_{n}(x) = \sum_{k=0}^{2^{n}} \exp(\frac{k}{2^{n}})\chi_{[\frac{k}{2^{n+1}},\frac{k+1}{2^{n+1}})} (x)$$ for \( n=1,2,\ldots \). The plot below is similar to the one already discussed for \(\sin(x)\).
For further information and results check this paper out.