Trivial measure

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.

Properties of the trivial measure

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.