Unit · year 3
MU-302 · Measure & Integration
Threads space · chance24 lectures5 theorems
The Lebesgue theory: a robust notion of size and integral.
PREREQUISITES
Lectures
| L01 | Why the Riemann Integral Is Not Enough — |
| L02 | Algebras, σ-Algebras, and Measurable Spaces — |
| L03 | Measures and Their Basic Properties — |
| L04 | Outer Measures |
| L05 | Carathéodory's Extension Theorem |
| L06 | Construction of Lebesgue Measure |
| L07 | Non-Measurable Sets and the Axiom of Choice — |
| L08 | Measurable Functions — |
| L09 | Simple Functions and Approximation — |
| L10 | The Lebesgue Integral of Non-Negative Functions — |
| L11 | The Monotone Convergence Theorem |
| L12 | Fatou's Lemma |
| L13 | Integrable Functions and Linearity — |
| L14 | The Dominated Convergence Theorem |
| L15 | Comparison with the Riemann Integral — |
| L16 | Modes of Convergence: a.e., in Measure, in Lᵖ — |
| L17 | Lᵖ Spaces and Hölder's Inequality — |
| L18 | Minkowski's Inequality and Completeness of Lᵖ — |
| L19 | Product Measures |
| L20 | The Fubini–Tonelli Theorem |
| L21 | Signed Measures and the Hahn Decomposition — |
| L22 | Radon–Nikodym Derivatives and Densities — |
| L23 | Differentiation of Measures and Lebesgue Points — |
| L24 | Synthesis: Limits That Pass Through the Integral |
Theorems in this unit
T-084
Carathéodory's extension theorem
A premeasure extends to a genuine measure.
T-085
The monotone convergence theorem
Integrals commute with increasing limits of non-negative functions.
T-086
Fatou's lemma
The integral of a liminf is at most the liminf of integrals.
T-087
The dominated convergence theorem
A dominated pointwise limit may be integrated term by term.
T-088
The Fubini–Tonelli theorem
When iterated integrals may be exchanged.