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.[1]
Properties of the trivial measure
Let μ denote the trivial measure on some measurable space (X, Σ).
A measure ν is the trivial measure μif and only ifν(X) = 0.
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 n-dimensional
Euclidean spaceRn with its usual σ-algebra and n-dimensional
Lebesgue measureλn, μ is a
singular measure with respect to λn: simply decompose Rn as A = Rn \ {0} and B = {0} and observe that μ(A) = λn(B) = 0.
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.[1]
Properties of the trivial measure
Let μ denote the trivial measure on some measurable space (X, Σ).
A measure ν is the trivial measure μif and only ifν(X) = 0.
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 n-dimensional
Euclidean spaceRn with its usual σ-algebra and n-dimensional
Lebesgue measureλn, μ is a
singular measure with respect to λn: simply decompose Rn as A = Rn \ {0} and B = {0} and observe that μ(A) = λn(B) = 0.