Measure Theory

Checkpoints

• Introduction to Measure Basis Measure theory generalises length, area, and volume into a single notion that can assign a size to far stranger sets. This checkpoint motivates the problem of measuring sets, explains why naïve length on the rationals already breaks down, and previews how σ-algebras and outer measure fix it.
• Lebesgue Measure Basis The Lebesgue measure is the outer measure built from interval lengths, restricted to its Carathéodory-measurable sets. This checkpoint constructs Lebesgue outer measure on ℝ, identifies the Lebesgue σ-algebra, and works out the basic computations — open sets, closed sets, null sets, and translation invariance.
• Carathéodory's Measurability Criterion Basis Carathéodory's criterion picks out the sets on which an outer measure becomes a true (countably additive) measure: E is measurable when it splits every test set additively. This checkpoint states the criterion, proves that the measurable sets form a σ-algebra, and shows that the restricted outer measure is a complete measure.
• Outer Measure Basis An outer measure assigns a size to every subset of a space by taking the infimum of total lengths over countable open covers. This checkpoint defines outer measure, proves its monotonicity and countable sub-additivity, and shows why countable additivity fails in general — motivating the restriction to measurable sets.
• σ-Algebra Basis A σ-algebra is a collection of subsets that is closed under complement and countable union — the natural domain on which a measure can be defined. This checkpoint states the axioms, works through the Borel σ-algebra as the canonical example, and explains why countable (rather than finite) closure is the right strength.