Measure and Integration

Why length needs a theory, and what integration becomes once it has one

About this course

Measure theory from the failures of the Riemann integral through outer measure, a non-measurable set, and Carathéodory's criterion for Lebesgue measure.

What you will be able to do

  • State precisely what the Riemann integral cannot do
  • Construct outer measure by covering sets with intervals
  • Explain why a non-measurable set must exist
  • Apply Carathéodory's criterion to define Lebesgue measure
  • Distinguish measurable sets from sets that carry no length

What this course covers

  • Carathéodory's Criterion for Measurability
  • Null Sets Are Measurable (Completeness)
  • Half-Lines Are Measurable (Interval-Cutting Argument)
  • Script M Is a Sigma-Algebra
  • Lebesgue Measure: Countable Additivity and Its Consequences
  • Continuity of Measure (From Below and Above)
  • The Vitali Set Lies Outside Script M
  • Riemann integral: partitions, upper and lower sums
  • Enumeration of the rationals on the unit interval
  • Increasing sequence converging to Dirichlet's function
  • Fat Cantor set construction and its length
  • Decreasing sequence converging to the indicator of C
  • Completeness failure of Riemann integrable functions
  • Wish list for a better integral: three promises and a duty
  • Outer measure: motivation and definition
  • Basic properties: empty set, points, and monotonicity
  • Open vs. closed intervals in the definition
  • Translation invariance of outer measure
  • Countable subadditivity and null sets
  • Outer measure of an interval equals its length
  • Uncountability of the reals via outer measure
  • Why countable covers outperform finite covers
  • Outer measure of the middle-thirds Cantor set
  • Outer measure of the fat Cantor set
  • Comparison of four subsets of the unit interval
  • Open question: is outer measure countably additive?
  • The Four Demands for a Length Function
  • Equivalence Classes of the Rationals on the Real Line
  • The Vitali Set and the Axiom of Choice
  • The Vitali Set Sandwich and the Non-Existence Theorem
  • Outer Measure Fails Countable and Finite Additivity on Vitali Sets
  • Properties of Vitali Sets and the Path Forward to Measurable Sets

Syllabus

  1. The Limits of the Riemann Integral

    1. L1 · What the Riemann Integral Cannot Do

  2. Outer Measure

    1. L2 · Outer Measure: Covering Sets by Intervals
    2. L3 · A Set No Length Can Measure

  3. Lebesgue Measure

    1. L4 · Carathéodory's Criterion and Lebesgue Measure