Trivial measure

In mathematics, specifically in measure theory, the trivial measure on any measurable space is the measure μ which assigns zero measure to every measurable set: μ = 0 for all A in Σ.

Properties of the trivial measure

Let μ denote the trivial measure on some measurable space.

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.

Suppose that X is a topological space and that Σ is the Borel σ-algebra on 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.