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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Dominated convergence theorem
An integrable dominating function permits exchanging almost-everywhere limits and integrals.
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Measurable functions
Inverse images of appropriate Borel sets must be measurable.
Monotone convergence theorem
Increasing nonnegative measurable functions allow exchanging limit and integral.
Almost everywhere
A property holds almost everywhere if its exceptional set has measure zero.
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.
Knowledge connections
- Monotone convergence theorem · Related · Map context
- Measurable functions · Related · Map context