Measure in mathematics

In mathematics, a measure is said to be **saturated** if every locally measurable set is also measurable.^{[1]} A set $E$, not necessarily measurable, is said to be a **locally measurable set** if for every measurable set $A$ of finite measure, $E\cap A$ is measurable. $\sigma$-finite measures and measures arising as the restriction of outer measures are saturated.

**^**Bogachev, Vladmir (2007).*Measure Theory Volume 2*. Springer. ISBN 978-3-540-34513-8.

Basic concepts | |||||
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Sets | |||||

Types of Measures | - Atomic
- Baire
- Banach
- Besov
- Borel
- Brown
- Complex
- Complete
- Content
- (Logarithmically) Convex
- Decomposable
- Discrete
- Equivalent
- Finite
- Inner
- (Quasi-) Invariant
- Locally finite
- Maximising
- Metric outer
- Outer
- Perfect
- Pre-measure
- (Sub-) Probability
- Projection-valued
- Radon
- Random
- Regular
- Saturated
- Set function
- σ-finite
- s-finite
- Signed
- Singular
- Spectral
- Strictly positive
- Tight
- Vector
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Particular measures | |||||

Maps | |||||

Main results | - Carathéodory's extension theorem
- Convergence theorems
- Decomposition theorems
- Egorov's
- Fatou's lemma
- Fubini's
- Hölder's inequality
- Minkowski inequality
- Radon–Nikodym
- Riesz–Markov–Kakutani representation theorem
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Other results |
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Applications & related |