In mathematics, specifically in measure theory, the trivial measure on any measurable space (X, Σ) is the measure μ which assigns zero measure to every measurable set: μ(A) = 0 for all A in Σ.[1]
Properties of the trivial measureedit
Let μ denote the trivial measure on some measurable space (X, Σ).
A measure ν is the trivial measure μif and only ifν(X) = 0.
If X is a Hausdorff topological space with its Borel σ-algebra, then μ trivially satisfies the condition to be a tight measure. Hence, μ is also a Radon measure. In fact, it is the vertex of the pointed cone of all non-negative Radon measures on X.
If X is n-dimensional Euclidean spaceRn with its usual σ-algebra and n-dimensional Lebesgue measureλn, μ is a singular measure with respect to λn: simply decompose Rn as A = Rn \ {0} and B = {0} and observe that μ(A) = λn(B) = 0.
Referencesedit
^Porter, Christopher P. (2015-04-01). "Trivial Measures are not so Trivial". Theory of Computing Systems. 56 (3): 487–512. arXiv:1503.06332. doi:10.1007/s00224-015-9614-8. ISSN 1433-0490.
January 01, 1970
trivial, measure, mathematics, specifically, measure, theory, trivial, measure, measurable, space, measure, which, assigns, zero, measure, every, measurable, properties, trivial, measure, editlet, denote, trivial, measure, some, measurable, space, measure, tri. In mathematics specifically in measure theory the trivial measure on any measurable space X S is the measure m which assigns zero measure to every measurable set m A 0 for all A in S 1 Properties of the trivial measure editLet m denote the trivial measure on some measurable space X S A measure n is the trivial measure m if and only if n X 0 m is an invariant measure and hence a quasi invariant measure for any measurable function f X X Suppose that X is a topological space and that S is the Borel s algebra on X m trivially satisfies the condition to be a regular measure m is never a strictly positive measure regardless of X S since every measurable set has zero measure Since m X 0 m is always a finite measure and hence a locally finite measure If X is a Hausdorff topological space with its Borel s algebra then m trivially satisfies the condition to be a tight measure Hence m is also a Radon measure In fact it is the vertex of the pointed cone of all non negative Radon measures on X If X is an infinite dimensional Banach space with its Borel s algebra then m is the only measure on X S that is locally finite and invariant under all translations of X See the article There is no infinite dimensional Lebesgue measure If X is n dimensional Euclidean space Rn with its usual s algebra and n dimensional Lebesgue measure ln m is a singular measure with respect to ln simply decompose Rn as A Rn 0 and B 0 and observe that m A ln B 0 References edit Porter Christopher P 2015 04 01 Trivial Measures are not so Trivial Theory of Computing Systems 56 3 487 512 arXiv 1503 06332 doi 10 1007 s00224 015 9614 8 ISSN 1433 0490 Retrieved from https en wikipedia org w index php title Trivial measure amp oldid 1169945064, wikipedia, wiki, book, books, library,