Measure and Integration
Explore measurable structures; functions and convergence; product spaces and densities. Concepts and connections are available independently of video coverage.
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Measurable functions
Inverse images of appropriate Borel sets must be measurable.
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Dominated convergence theorem
An integrable dominating function permits exchanging almost-everywhere limits and integrals.
Fubini and Tonelli theorems
Product integrals can be iterated under integrability or nonnegativity and measure-space hypotheses.
Lebesgue integral
Integration via simple-function approximation handles measurable functions beyond Riemann integration.
Sigma-algebras
A sigma-algebra is closed under complements and countable unions.
Almost everywhere
A property holds almost everywhere if its exceptional set has measure zero.
Knowledge connections
- Lebesgue integral · Related · Map context
- Dominated convergence theorem · Related · Map context
- Sigma-algebras · Related · Map context
- Fubini and Tonelli theorems · Related · Map context
- Lebesgue integral · Next concept · Map context