Here is a way to prove one direction of Lebesgue’s theorem on Riemann integrable functions. (1) For each n ≥ 1 and each   be a point in   be the probability measure that assigns mass 1/n to each point...

Here is a way to prove one direction of Lebesgue’s theorem on Riemann integrable functions. (1) For each n ≥ 1 and each
  be a point in
  be the probability measure that assigns mass 1/n to each point
  Show that Pn converges weakly to P, where P is a Lebesgue measure on [0, 1].

(2) Suppose f is a bounded function which is continuous at almost every point of [0, 1]. Show that

  Note that
 is a Riemann sum approximation to is a Riemann sum approximation to

Chapter 32