The Rest of Measure Theory
Advanced Topics, Abstract Spaces, and Further Research
The topics covered so far are not an exhaustive treatment of measure theory, rather some of the most notable results. Most abstract measure theory becomes quite intuitive after a solid grounding in Lebesgue theory. Below is a roadmap of essential advanced topics.
Core Advanced Theorems
Fubini's & Tonelli's Theorems
Conditions under which the order of integration can be switched in iterated integrals.
Lp Theory
Exploring function spaces, Hölder's Inequality, and Young's Inequality for convolutions.
Abstract Measure Theory
Transitioning from the Euclidean plane to general spaces allows for a much broader application of these tools in probability and functional analysis.
- Foundations: Measure Spaces, Metric Measure Spaces, and \(\sigma\)-finiteness.
- Constructing Measures: Premeasures, Algebras, and the Carathéodory Criterion.
- Comparing Measures: Signed Measures, Absolute Continuity, and Mutually Singular Measures.
- The Radon–Nikodym Theorem: A fundamental result relating two different measures on the same space.
Functional Analysis & Beyond
The study of Hilbert Spaces generalizes the notion of Euclidean space to infinite dimensions, providing the mathematical framework for Quantum Mechanics and Signal Processing.