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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Radon–Nikodym theorem
Under suitable sigma-finiteness, absolute continuity of measures yields an integral density.
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Fubini and Tonelli theorems
Product integrals can be iterated under integrability or nonnegativity and measure-space hypotheses.
Lp spaces
Integrable powers define function spaces modulo equality almost everywhere.
Almost everywhere
A property holds almost everywhere if its exceptional set has measure zero.
Dominated convergence theorem
An integrable dominating function permits exchanging almost-everywhere limits and integrals.
Lebesgue integral
Integration via simple-function approximation handles measurable functions beyond Riemann integration.
Knowledge connections
- Lp spaces · Related · Map context
- Fubini and Tonelli theorems · Related · Map context