In mathematics, a pre-measure is a set function that is, in some sense, a precursor to a bona fide measure on a given space. Indeed, one of the fundamental theorems in measure theory states that a pre-measure can be extended to a measure.
Definition
Families of sets over
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Is necessarily true of ![{\displaystyle {\mathcal {F))\colon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c806bc7022198fb7b8ddd4a0b376329bb77e00c) or, is closed under:
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Directed by
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F.I.P.
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π-system
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![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
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Semiring
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![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
Never
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Semialgebra (Semifield)
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![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
Never
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Monotone class
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![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
only if ![{\displaystyle A_{i}\searrow }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba4f0f9c907ac9321bf8494f69cc190cbf8a56d) |
only if ![{\displaystyle A_{i}\nearrow }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b851ff0dcb2264bbedafbef85a71e4f98c842865) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
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𝜆-system (Dynkin System)
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![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
only if
![{\displaystyle A\subseteq B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b09068bd2f7ba899aeb883ebe670b2ad07b0c851) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
only if or they are disjoint |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
Never
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Ring (Order theory)
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![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
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Ring (Measure theory)
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![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
Never
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δ-Ring
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![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
Never
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𝜎-Ring
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![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
Never
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Algebra (Field)
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![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
Never
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𝜎-Algebra (𝜎-Field)
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![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
Never
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Dual ideal
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![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
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Filter
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![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
Never |
Never |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![{\displaystyle \varnothing \not \in {\mathcal {F))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6c99d2db231b6a5af19206e95ff6d98d3019e9b) |
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Prefilter (Filter base)
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![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
Never |
Never |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![{\displaystyle \varnothing \not \in {\mathcal {F))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6c99d2db231b6a5af19206e95ff6d98d3019e9b) |
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Filter subbase
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![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
Never |
Never |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![{\displaystyle \varnothing \not \in {\mathcal {F))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6c99d2db231b6a5af19206e95ff6d98d3019e9b) |
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Open Topology
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![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![](https://upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png) (even arbitrary ) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
Never
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Closed Topology
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![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![](https://upload.wikimedia.org/wikipedia/commons/thumb/0/03/Green_check.svg/13px-Green_check.svg.png) (even arbitrary ) |
![No](https://upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
![Yes](https://upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png) |
Never
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Is necessarily true of ![{\displaystyle {\mathcal {F))\colon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c806bc7022198fb7b8ddd4a0b376329bb77e00c) or, is closed under:
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directed downward
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finite intersections
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finite unions
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relative complements
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complements in
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countable intersections
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countable unions
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contains
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contains
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Finite Intersection Property
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Additionally, a semiring is a π-system where every complement is equal to a finite disjoint union of sets in ![{\displaystyle {\mathcal {F)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e1656ae73ede684468b360e948a8a38e6e2c461)
A semialgebra is a semiring where every complement is equal to a finite disjoint union of sets in ![{\displaystyle {\mathcal {F)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e1656ae73ede684468b360e948a8a38e6e2c461)
are arbitrary elements of and it is assumed that ![{\displaystyle {\mathcal {F))\neq \varnothing .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ed685bdf4c75742b28ccec093cae48329c1a9d6)
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Let
be a ring of subsets (closed under union and relative complement) of a fixed set
and let
be a set function.
is called a pre-measure if
![{\displaystyle \mu _{0}(\varnothing )=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c99a758a9e120fcb7f015cf93775f9993af903a6)
and, for every countable (or finite) sequence
of pairwise disjoint sets whose union lies in
![{\displaystyle \mu _{0}\left(\bigcup _{n=1}^{\infty }A_{n}\right)=\sum _{n=1}^{\infty }\mu _{0}(A_{n}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d2faaa1cfcd8af47e69d15a307cab4a92f8cf20)
The second property is called
-additivity.
Thus, what is missing for a pre-measure to be a measure is that it is not necessarily defined on a sigma-algebra (or a sigma-ring).
Carathéodory's extension theorem
It turns out that pre-measures give rise quite naturally to outer measures, which are defined for all subsets of the space
More precisely, if
is a pre-measure defined on a ring of subsets
of the space
then the set function
defined by
![{\displaystyle \mu ^{*}(S)=\inf \left\{\left.\sum _{i=1}^{\infty }\mu _{0}(A_{i})\right|A_{i}\in R,S\subseteq \bigcup _{i=1}^{\infty }A_{i}\right\))](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb376ff1bdd81e2643deef8b5d6fe1ae60275e5a)
is an outer measure on
and the measure
induced by
on the
-algebra
of Carathéodory-measurable sets satisfies
for
(in particular,
includes
). The infimum of the empty set is taken to be
(Note that there is some variation in the terminology used in the literature. For example, Rogers (1998) uses "measure" where this article uses the term "outer measure". Outer measures are not, in general, measures, since they may fail to be
-additive.)