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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.

## Properties of the trivial measure

Let *μ* denote the trivial measure on some measurable space (*X*, Σ).

- A measure
*ν*is the trivial measure*μ*if and only if*ν*(*X*) = 0. *μ*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 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 (*X*, Σ), since every measurable set has zero measure.- Since
*μ*(*X*) = 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 (*X*, Σ) 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**R**^{n}with its usual*σ*-algebra and*n*-dimensional Lebesgue measure*λ*^{n},*μ*is a singular measure with respect to*λ*^{n}: simply decompose**R**^{n}as*A*=**R**^{n}\ {0} and*B*= {0} and observe that*μ*(*A*) =*λ*^{n}(*B*) = 0.