Measure Theory

Measure theory is the study of measures on a set, which are assignments of a “size” to some of its subsets. Measures give rise to the theory of Lebesgue integration, which is more general and has better properties than Riemann integration.

These notes closely follow the graduate course MA5205 Graduate Analysis I, offered in AY2018/19 Semester 1 by Prof Denny Leung.

• Measure Theory (I): $\sigma$-algebras
• Measure Theory (II): Measures
• Measure Theory (III): Measurable Functions
• Measure Theory (IV): Lebesgue Integration (I)
• Measure Theory (V): Lebesgue Integration (II)
• Measure Theory (VI): Convergence Theorems
• Measure Theory (VII): The Carathéodory Construction of Measures
• Measure Theory (VIII): The Lebesgue Measure on $\bb R^n$
• Measure Theory (IX): Littlewood's Three Principles
• Measure Theory (X): The Riemann-Lebesgue Theorem
• Measure Theory (XI): $L^p$ Spaces
• Measure Theory (XII): Product Measures
• Measure Theory (XIII): Convolution on $L^p(\bb R)$
• Measure Theory (XIV): Approximate Identities
• Measure Theory (XV): The Fourier Transform on $\bb R$
• Measure Theory (XVI): Vitali's Covering Theorem
• Measure Theory (XVII): Functions of Bounded Variation
• Measure Theory (XVIII): The Lebesgue Differentiation Theorem

(In progress, last updated 29 Dec 2018)