The Issue with Riemann Integral

This lecture discusses the problem with the Riemann integral in the context of Hilbert spaces, focusing on L²(RN, µ). It explores the incompleteness of the space Cc(RN, K) with the scalar product (f,g) = √RN f(x)g(x)dNxn, leading to the construction of a sequence of compactly supported functions that form a Cauchy sequence for the L2 norm but whose limit function is not Riemann integrable. The lecture demonstrates how the sequence converges simply to a function foo, with its integral value suggesting 1/2, highlighting the challenges of Riemann integrability and the limitations of finite interval partitions.