Measure and Integration Theory
Read this topicExplanations 7
- Applications of Lebesgue density theorem: MSE: arithmetic-sequence-in-a-lebesgue-measurable-set
By Victor Wang
- See Terry Tao's list of problem solving strategies for his class 245A in Fall 2010.
By Victor Wang
- Proving sets or functions are measurable For example, see MSE: Set of points at which sequence of measurable functions converge (another approach). Or see MSE: Set of continuity…
By Victor Wang
- List of useful examples Cantor, fat Cantor, etc. A strictly increasing continuous function on [0,1][0,1][0,1] may map a set of positive measure onto a set of measure zero. But i…
By Victor Wang
- Overview of measure theory A good reference for everything here is Royden-Fitzpatrick, 4th Ed., e.g. Section 3.3 on Littlewood's Three Principles. Role of countable additivity, …
By Victor Wang, Alex Federspiel
- Sequences of functions & modes of convergence Converges pointwise Converges pointwise a.e. (almost everywhere) Converges uniformly Converges almost uniformly Converges in LpL^pL…
By Victor Wang
- Overview of integration theory A good reference for everything here is Royden-Fitzpatrick, 4th Ed., e.g. Chapters 4-6. Riemann integral is modeled off of lower and upper step fu…
By Victor Wang
Problems 3
- Which of the following two statements is always true? I. If {Bn:n≥1}\{B_n:n\geq1\}{Bn:n≥1} increases to (its union) BBB, then limn→∞μ(Bn)=μ(B)\lim_{n\to\infty} \mu(B_n) = \mu(…
- Assuming you know that m∗([a,b])≥b−am^\ast([a,b]) \geq b-am∗([a,b])≥b−a for closed intervals [a,b][a,b][a,b] (with a≤ba\leq ba≤b), how can you show that m∗((0,1))≥1m^\ast((0,1))…
- Let f :R→Rf\colon \mathbb{R} \to \mathbb{R}f:R→R be a continuous function. In what sense is fff necessarily a measurable function? Select all that apply.