Measure and Integration Theory

See Terry Tao's list of problem solving strategies for his class 245A in Fall 2010.

Applications of Lebesgue density theorem:

• MSE: arithmetic-sequence-in-a-lebesgue-measurable-set

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 points of a real function.

For functions:

• uniform limits of continuous functions (see e.g. Wikipedia)
• pointwise a.e. limits of measurable functions
• min, max of finite collections/sequences
• inf, sup, lim inf, lim sup of countable sequences
• sums and products (maybe with some finiteness hypotheses)

List of useful examples

• Cantor, fat Cantor, etc.

• A strictly increasing continuous function on $[0,1]$ may map a set of positive measure onto a set of measure zero. But if its inverse is Lipschitz, then this cannot happen: only measure zero sets can map onto measure zero sets. See MSE: A Lipschitz transform maps measurable set to measurable.

• A continuous function may not pull Lebesgue measurable sets to Lebesgue measurable sets. But if it has a Lipschitz inverse, then it does. See MSE: A Lipschitz transform maps measurable set to measurable.

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, unions, and intersections (e.g. Zeno's paradox; also, intuition for why $G_\delta$ and $F_\sigma$ sets can be "measured", i.e. part of the consistent collection of Lebesgue measurable sets)

• Littlewood's first principle

Continuous, Riemann integrable, and Lebesgue measurable functions

• Riemann integral is still better for computation in many (most?) cases.

• Philosophical flexibility of Lebesgue measurable functions (esp. in context of integration of, say, $\chi_{ \mathbb{Q}}$ or other things that Riemann doesn't handle)

• Pointwise limits, Egoroff's theorem, and Littlewood's third principle

• Simple approximation of Lebesgue measurable functions, Lusin's theorem, and Littlewood's second principle

Sequences of functions & modes of convergence

• Converges pointwise
• Converges pointwise a.e. (almost everywhere)
• Converges uniformly
• Converges almost uniformly
• Converges in $L^p$-norm (see here for how to interpret $p=\infty$ case: it ends up being equivalent to "uniformly almost everywhere")
• Converges in measure

References

• Royden & Fitzpatrick, 4th Ed.

• John D. Cook: math diagrams, including one on modes of convergence

• Wikipedia: modes of convergence and annotated index

• Terry Tao: quiz and notes from 245A in Fall 2010

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 functions
• Lebesgue integral is modeled off of simple functions instead
• Riemann is bad for Dirichlet's function $\chi_{ \mathbb{Q}}$, the indicator function of the rational numbers; Lebesgue handles it better (roughly speaking, Riemann requires function preimages to have reasonable topology, requiring "simple vertical slabs" from the viewpoint of inputs, i.e. the independent variable, or requiring "interval-union-like horizontal slabs" from the viewpoint of outputs, i.e. the dependent variable; Lebesgue cares only about measurability of preimages, allowing "more complicated but still measurable horizontal slabs" from the viewpoint of outputs)
• See the interesting discussion at MO: Why should one still teach Riemann integration? as well.