Foundations of Advanced Analysis - Mod. A
Master's Degree in Mathematics
University of Messina
Academic Year 2025/2026
This course develops advanced topics in measure theory and integration, with particular emphasis on integration on measure spaces, product measures, Lp spaces, modes of convergence, absolute continuity of measures, and Vitali's theorem.
Course Contents
1. Integration on a Measure Space (9 hours)
Simple functions. Approximation of non-negative measurable real-valued functions by simple functions. Integral of simple functions. Properties of the integral of simple functions. Properties holding almost everywhere. Integral of everywhere-defined or almost everywhere-defined non-negative measurable functions. Properties of the integral. Beppo Levi's Monotone Convergence Theorem. Integration of series. Distributive property of the integral with respect to the measure. Measures on ℘(M), where M is a countable set. μ-quasi-integrable functions and their integral. μ-integrable functions. Conditions for μ-integrability. Properties of the integral. Examples of integration. Integral over a subset of Ω. Signed measures with density with respect to a measure μ. Properties of the signed measure fμ. Transformation of integrals from one measure space to another.
2. Integration with Respect to a Product Measure (8 hours)
Product of σ-algebras. Generators of the product σ-algebra. Product of σ-finite measures. Tonelli's Theorem. Fubini's Theorem. Tonelli's and Fubini's Theorems for the Lebesgue measure mp+q.
3. The Spaces Lp(μ) – Part I (4 hours)
μ-measurable p-integrable functions. Hölder's and Minkowski's inequalities. Clarkson's inequalities. The spaces Lp(μ), 0 < p < +∞. The space L∞(μ). Completeness of L∞(μ).
4. Various Modes of Convergence of Sequences of Measurable Real-Valued Functions (8 hours)
Fatou's Lemma and some of its consequences. Almost everywhere convergence. Convergence in the mean of order p. Dominated Convergence Theorem. Completeness of Lp(μ), 0 < p < +∞. Relations between convergence in the mean and almost everywhere convergence. Quasi-uniform convergence. Characterization of quasi-uniform convergence. Severini–Egorov Theorem. Convergence in measure. Weyl–Riesz criterion for convergence in measure. Summary of the relationships among the various notions of convergence.
5. The Spaces Lp(μ) – Part II (9 hours)
Inclusions among Lp spaces. Characterization of the inclusion Lp(μ) ⊆ Lq(μ). Topological dual of Lp spaces for 1 ≤ p ≤ +∞. Reflexivity of Lp spaces for 1 ≤ p ≤ +∞. Density theorems in Lp spaces. Separability of Lp spaces for 1 ≤ p ≤ +∞.
6. Measures with Density (6 hours)
Signed measures with density with respect to a measure μ. Integration with respect to a measure with density. Signed measures absolutely continuous with respect to a measure μ. Absolute continuity in the sense of Vitali and in the sense of Caccioppoli. Comparison between the different notions of absolute continuity. Radon–Nikodym Theorem. Uniqueness of the density.
7. Characterization of Convergence in the Mean: Vitali's Theorem (4 hours)
Equi-absolute continuity in the sense of Vitali and in the sense of Caccioppoli. Vitali's Theorem, together with its variants and consequences. Characterization of convergence in the mean through the convergence of norms.