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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Monotone convergence theorem
Increasing nonnegative measurable functions allow exchanging limit and integral.
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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.
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.
Lebesgue measure
Lebesgue measure extends interval length to a complete measure on a class of real sets.
Knowledge connections
- Almost everywhere · Related · Map context
- Dominated convergence theorem · Related · Map context