Trivial measure

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 an 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.