The rigorous foundation of modern analysis through 24 interactive demonstrations. From sigma-algebras to Lp spaces, with the Cantor set and convergence theorems along the way.
See also: Real Analysis for the Riemann integral that Lebesgue measure extends, Probability for probability as integration against a measure, and Fourier Analysis for L² spaces and Plancherel's theorem.
From set systems to σ-algebras, Borel sets, and the structure that makes measurement possible
Construct the Lebesgue measure from outer measure via Carathéodory's criterion
Build the Cantor set, explore its paradoxes, and animate the devil's staircase
Why the Vitali set defeats any translation-invariant measure on the reals
Simple function approximation, Littlewood's three principles, and Lusin's theorem
Build the integral from simple functions and see why it dominates Riemann's approach
MCT, Fatou's lemma, and DCT — the power tools of modern analysis
Function spaces, norms, completeness, and the Hölder and Minkowski inequalities