· 5 min
Residues and Contour Integration
Evaluating integrals by what a function leaves behind at its poles. The Laurent series and the residue, the residue theorem, and a contour computation of the Cauchy characteristic function.
#integration
· 5 min
Uniform Integrability and the Vitali Theorem
The exact condition for convergence in the mean. Uniform integrability, its characterisation by uniform absolute continuity, and the Vitali theorem that convergence in measure upgrades to L-one convergence exactly when the sequence is uniformly integrable.
· 5 min
The L-p Spaces
The Banach spaces of integrable powers. Young's inequality, the Holder and Minkowski inequalities that make the p-norm a norm, and the completeness theorem that promotes every L-p to a Banach space, the family of which L-squared is the one Hilbert member.
· 6 min
The Riemann Integral
Area as a limit of sums squeezed between over and under estimates. Upper and lower sums, the Riemann criterion for integrability, the integrability of continuous functions through uniform continuity, and the fundamental theorem of calculus tying the integral to the derivative.
· 7 min
Measures and Integration
The Lebesgue integral built from simple functions, and the three convergence theorems that make it usable. We prove monotone convergence, deduce Fatou and dominated convergence, and record the L^p inequalities.
· 7 min
Product Measures and Fubini's Theorem
When a double integral equals an iterated one. The product sigma-algebra and the measurability of sections, the construction of the product measure, and the Tonelli and Fubini theorems that exchange the order of integration for nonnegative and for integrable functions.