Trivial measure
Appearance
Inmathematics,specifically inmeasure theory,thetrivial measureon anymeasurable space(X,Σ) is the measureμwhich assigns zero measure to every measurable set:μ(A) = 0 for allAin Σ.[1]
Properties of the trivial measure
[edit]Letμdenote the trivial measure on some measurable space (X,Σ).
- A measureνis the trivial measureμif and only ifν(X) = 0.
- μis aninvariant measure(and hence aquasi-invariant measure) for anymeasurable functionf:X→X.
Suppose thatXis atopological spaceand that Σ is theBorelσ-algebraonX.
- μtrivially satisfies the condition to be aregular measure.
- μis never astrictly positive measure,regardless of (X,Σ), since every measurable set has zero measure.
- Sinceμ(X) = 0,μis always a finite measure, and hence alocally finite measure.
- IfXis aHausdorfftopological space with its Borelσ-algebra, thenμtrivially satisfies the condition to be atight measure.Hence,μis also aRadon measure.In fact, it is the vertex of thepointed coneof all non-negative Radon measures onX.
- IfXis aninfinite-dimensionalBanach spacewith its Borelσ-algebra, thenμis the only measure on (X,Σ) that is locally finite and invariant under all translations ofX.See the articleThere is no infinite-dimensional Lebesgue measure.
- IfXisn-dimensionalEuclidean spaceRnwith its usualσ-algebra andn-dimensionalLebesgue measureλn,μis asingular measurewith respect toλn:simply decomposeRnasA=Rn\ {0} andB= {0} and observe thatμ(A) =λn(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.ISSN1433-0490.