Trivial measure

Properties of the trivial measure

• A measure ν is the trivial measure μ if and only if ν = 0.
• μ is an invariant measure for any measurable function f : X → X.

• μ trivially satisfies the condition to be a regular measure.
• μ is never a strictly positive measure, regardless of, since every measurable set has zero measure.
• Since μ = 0, μ is always a finite measure, and hence a locally finite measure.
• 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 an infinite-dimensional Banach space with its Borel σ-algebra, then μ is the only measure on 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 σ-algebra and n-dimensional Lebesgue measure λⁿ, μ is a singular measure with respect to λⁿ: simply decompose Rⁿ as A = Rⁿ \ and B = and observe that μ = λⁿ = 0.