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Euler measure

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In measure theory, the Euler measure of a polyhedral set equals the Euler integral of its indicator function.

The magnitude of an Euler measure

By induction, it is easy to show that independent of dimension, the Euler measure of a closed bounded convex polyhedron always equals 1, while the Euler measure of a d-D relative-open bounded convex polyhedron is $(-1)^{d}$ .

See also

Measure theory

Notes

Weisstein, Eric W. "Euler Measure". Wolfram MathWorld. Retrieved 7 July 2018.

External links

Exponentiation and Euler measure

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Basic concepts

Sets

Types of measures

Particular measures

Maps

Measurable function
- Bochner
- Strongly
- Weakly
Convergence: almost everywhere
of measures
in measure
of random variables
- in distribution
- in probability
Cylinder set measure
Random: compact set
element
measure
process
variable
vector
Projection-valued measure

Carathéodory's extension theorem
Convergence theorems
Decomposition theorems
- Hahn
- Jordan
- Maharam's
Egorov's
Fatou's lemma
Fubini's
- Fubini–Tonelli
Hölder's inequality
Minkowski inequality
Radon–Nikodym
Riesz–Markov–Kakutani representation theorem

Other results

Disintegration theorem Lifting theory Lebesgue's density theorem Lebesgue differentiation theorem Sard's theorem Vitali–Hahn–Saks theorem
For Lebesgue measure	Isoperimetric inequality Brunn–Minkowski theorem Milman's reverse Minkowski–Steiner formula Prékopa–Leindler inequality Vitale's random Brunn–Minkowski inequality

Applications & related

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