Measure and integration Construction of measures Concept
Outer measure
Summary
An outer measure assigns a size in to every subset of a set, not only to the sets of a -algebra. It is monotone and countably subadditive, but in general not additive. Through Carathéodory's criterion every outer measure produces a -algebra on which it is a complete measure. This is the standard way to build Lebesgue measure and Hausdorff measures.
Where this leads
31 pages build on this one.
- Nonexistence of infinite-dimensional Lebesgue measure
- Lebesgue measure
- Standard probability space
- Minkowski content
- Hardy–Littlewood maximal function
- Brunn–Minkowski theorem
- Liouville's theorem on phase-space volume
- Hausdorff measure
- Lebesgue point
- Littlewood's three principles
- Duistermaat–Heckman formula
- Hahn–Kolmogorov theorem
- Lusin's theorem
- Layer cake representation
- Smith–Volterra–Cantor set
- Carathéodory's criterion
- Young's convolution inequality
- Banach measure problem
- Calderón–Zygmund lemma
- Isoperimetric inequality
- Gaussian measure
- Coarea formula
- Packing measure
- Fundamental theorem of calculus for Lebesgue integrals
- Lebesgue differentiation theorem
- Measure problem of Lebesgue
- Lebesgue's density theorem
- Riemann–Lebesgue lemma
- Vitali set
- Metric outer measure
- Isomorphism theorem for measure algebras
Further reading
About this page
| Tag | BJ314H17HC |
|---|---|
| URI | m2://BJ314H17HC |
| Kind | concept |
| Writing | W0 stub |
| MSC | 28A12 |
| Same as | |
| Changed | 2 November 2026 |
Markdown · JSON · Edit on GitHub
What this page needs for W1
- a source with a locator (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.
What links here
Used by
- Lebesgue measure
- Hausdorff measure
- Hahn–Kolmogorov theorem
- Carathéodory's criterion
- Packing measure
- Metric outer measure