Report for MA502 Presentation: Lebesgue’s Criterion for Riemann Integrability

We covered Riemann integrals in the first three weeks in MA502 this semester (Chapter 11 in [1]). This report explores a necessary and sufficient condition for determining Riemann integrability of f(x) solely from its properties. This condition is known as Lebesgue’s criterion and elucidating the proof of this condition is the aim of this report. This report revisits the concepts learnt in MA501/502 in Section 2. Section 2 also covers an example showing that functions discontinuous everywhere need not be Riemann integrable. Towards the end, this section reminds the readers a few results from measure theory useful in proving Lebesgue’s criterion for Riemann integrability. Section 3 states the Lebesgue’s criterion and provides examples of functions with countably infinite and uncountably infinite discontinuities which are Riemann integrable to motivate the usefulness of this criterion. Finally, Section 4 provides the proof for the Lebesgue’s criterion for Riemann integrability. The references used in this report are attached in the end.