Notes on book "Measure Theory (2nd ed.) - Cohn Donald L."

Measure Thoery: Introduction

The Riemann Integral -- Riemann's Definition

A tagged partition of an interval is a partion of , together with a sequence of numbers such that holds for . The mesh of a partion is defined by . The Riemann sum corresponding to the function and the tagged partition is defined by

Then, according to Riemann's definition, the function is integrable over if there is a number such that

where the limit is taken as the mesh of approaches 0. If we express this in terms of 's and 's, we see that the function is Riemann integrable, with integral , if for every positive , there is a positive such that holds for each tagged partition of that satisfies .

Limitation of Riemann Integral

Riemann Integral cannot handle point-wise convergence case, or Diltac function. While, Lebesgue Integral can do those.

References

[1] @book{cohn2013measure, title={Measure theory}, author={Cohn, Donald L}, year={2013}, publisher={Springer} }