Difference between revisions of "Outer-measure"
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==For every [[pre-measure]]== | ==For every [[pre-measure]]== | ||
<math>\mu^*=\text{Inf}\left.\left\{\sum^\infty_{n=1}\bar{\mu(E_n)}\right\vert E_n\in R\ \forall n,\ E\subset\bigcup^\infty_{n=1}E_n\right\}</math> is an outer measure. | <math>\mu^*=\text{Inf}\left.\left\{\sum^\infty_{n=1}\bar{\mu(E_n)}\right\vert E_n\in R\ \forall n,\ E\subset\bigcup^\infty_{n=1}E_n\right\}</math> is an outer measure. | ||
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Definition
An outer-measure, μ∗ is a set function from a hereditary σ-ring, H, to the (positive) extended real values, ˉR≥0, that is[1]:
- ∀A∈H[μ∗(A)≥0] - non-negative
- ∀A,B∈H[A⊆B⟹μ∗(A)≤μ∗(B)] - monotonic
- ∀(An)∞n=1⊆H[μ∗(⋃∞n=1An)≤∑∞n=1μ∗(An)] - countably subadditive
In words, μ∗ is:
- an extended real valued countably subadditive set function that is monotonic and non-negative with the property: μ∗(∅)=0 defined on a hereditary σ-ring
For every pre-measure
μ∗=Inf{∞∑n=1¯μ(En)|En∈R ∀n, E⊂∞⋃n=1En} is an outer measure.
References
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