Measure Theory

Syllabus

• Surfaces and surface integrals: surface parametrization, normals, flux; surface/area integrals.
• Divergence and curl; Newton–Leibniz, Gauss–Ostrogradsky (divergence) and Stokes theorems.
• Set systems and σ-algebras; Borel sets; measurable spaces; measures and signed measures.
• Measure extension theorem (existence/uniqueness).
• Lebesgue and Lebesgue–Stieltjes measures; regularity; examples of non-measurable sets (conceptual).
• Integration with respect to a measure: measurable functions; integrals of nonnegative, real and complex-valued functions; algebraic properties.
• Integrals of function sequences; convergence theorems (monotone convergence / Beppo Levi, dominated convergence / Lebesgue), Fatou lemma; almost everywhere convergence.
• Riemann vs Lebesgue–Stieltjes integration; lower/upper envelopes; product measures; Fubini theorem (statement).
• Signed and complex measures: total variation, norm; Hahn and Jordan decompositions; weak-* convergence of Borel measures.
• Absolute continuity and singularity; Lebesgue decomposition; Radon–Nikodym theorem.
• Change of variables / substitution for measure integrals; integration w.r.t. signed/complex measures.
• Functions of bounded variation; absolutely continuous vs singular functions; Cantor function; derivatives.
• Real and complex Lp spaces; Hölder, Cauchy–Schwarz, Minkowski inequalities; Riesz–Fischer theorem.