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Measure and Integration
Explore measurable structures; functions and convergence; product spaces and densities. Concepts and connections are available independently of video coverage.
Prerequisites and connections
Sets
A set is a collection of distinct objects; membership is different from inclusion.
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Measurable structures
Sigma-algebras
A sigma-algebra is closed under complements and countable unions.
Video explanations to be addedMeasures
A measure assigns nonnegative size and is countably additive on disjoint measurable sets.
Video explanations to be addedOuter measures
An outer measure is monotone and countably subadditive on all subsets.
Video explanations to be addedLebesgue measure
Lebesgue measure extends interval length to a complete measure on a class of real sets.
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Functions and convergence
Measurable functions
Inverse images of appropriate Borel sets must be measurable.
Video explanations to be addedLebesgue integral
Integration via simple-function approximation handles measurable functions beyond Riemann integration.
Video explanations to be addedAlmost everywhere
A property holds almost everywhere if its exceptional set has measure zero.
Video explanations to be addedMonotone convergence theorem
Increasing nonnegative measurable functions allow exchanging limit and integral.
Video explanations to be addedDominated convergence theorem
An integrable dominating function permits exchanging almost-everywhere limits and integrals.
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Product spaces and densities
Fubini and Tonelli theorems
Product integrals can be iterated under integrability or nonnegativity and measure-space hypotheses.
Video explanations to be addedLp spaces
Integrable powers define function spaces modulo equality almost everywhere.
Video explanations to be addedRadon–Nikodym theorem
Under suitable sigma-finiteness, absolute continuity of measures yields an integral density.
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