Measure Theory and Integration
Year 3 · 5 sections · 46 concepts · 76 practice questions
What's covered
free1 · Sigma-algebras and measures8 concepts
- Overview 1 · The plan of this section
- Definition 1.1 · σ-algebras
- Definition 1.2 · Generated σ-algebras and Borel sets
- Definition 1.3 · Measures
- Example 1.4 · Examples of measures
- Proposition 1.5 · Basic properties of measures
- Definition 1.6 · Null sets and almost everywhere
- Check 1.7 · Check: verifying a measure
2 · Lebesgue measure10 concepts
- Overview 2 · The plan of this section
- Definition 2.1 · Lebesgue outer measure
- Proposition 2.2 · Properties of outer measure
- Proposition 2.3 · The outer measure of an interval is its length
- Definition 2.4 · Carathéodory measurable sets
- Theorem 2.5 · Carathéodory's theorem
- Theorem 2.6 · Lebesgue measure
- Example 2.7 · The Cantor set is an uncountable null set
- Theorem 2.8 · A non-measurable set (Vitali)
- Check 2.9 · Check: computing Lebesgue measures
3 · Measurable functions and the integral9 concepts
- Overview 3 · The plan of this section
- Definition 3.1 · Measurable functions
- Proposition 3.2 · Operations on measurable functions
- Definition 3.3 · Simple functions
- Proposition 3.4 · Approximation by simple functions
- Definition 3.5 · The integral of a non-negative function
- Definition 3.6 · Integrable functions
- Theorem 3.7 · Riemann integrable functions are Lebesgue integrable
- Check 3.8 · Check: computing Lebesgue integrals
4 · Convergence theorems9 concepts
- Overview 4 · The plan of this section
- Theorem 4.1 · The monotone convergence theorem
- Corollary 4.2 · Linearity and monotonicity of the integral
- Lemma 4.3 · Fatou's lemma
- Theorem 4.4 · The dominated convergence theorem
- Corollary 4.5 · Integrating series term by term
- Proposition 4.6 · Continuity and differentiation under the integral
- Example 4.7 · When the hypotheses fail
- Check 4.8 · Check: justifying limit interchanges
5 · Product measures and Lp spaces10 concepts
- Overview 5 · The plan of this section
- Definition 5.1 · Product σ-algebras and measures
- Theorem 5.2 · Tonelli's theorem
- Theorem 5.3 · Fubini's theorem
- Example 5.4 · A failure of Fubini without integrability
- Definition 5.5 · L^p spaces
- Theorem 5.6 · Hölder's inequality
- Corollary 5.7 · Minkowski's inequality
- Theorem 5.8 · The Riesz–Fischer theorem
- Check 5.9 · Check: swapping integrals and using Lp
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Proposition 1.5 · Basic properties of measures
Monotone, countably subadditive, continuous from below; from above if finite.
Proof
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