The Integral That Changed Math Forever

Abide by Reason uses the Dirichlet function to illustrate for analysis students how Riemann integration can break down under limits of integrable functions.

This note targets learners transitioning from Riemann’s “rectangle under the curve” picture to Lebesgue’s measure-theoretic approach.

Abide by Reason works out concrete Lebesgue integrals for viewers who want to see how measure-based reasoning handles functions that defeat Riemann integration.

This note is for learners ready to see how Lebesgue integrals are built systematically from simple building blocks and why monotone convergence is central.

Abide by Reason offers an analogy for viewers wondering why Lebesgue integration is worth the extra abstraction beyond Riemann’s classical construction.