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Energy of measures on compact Riemannian manifolds

Kathryn E. Hare, Maria Roginskaya (2003)

Studia Mathematica

We investigate the energy of measures (both positive and signed) on compact Riemannian manifolds. A formula is given relating the energy integral of a positive measure with the projections of the measure onto the eigenspaces of the Laplacian. This formula is analogous to the classical formula comparing the energy of a measure in Euclidean space with a weighted L² norm of its Fourier transform. We show that the boundedness of a modified energy integral for signed measures gives bounds on the Hausdorff...

Existence and integral representation of regular extensions of measures

Werner Rinkewitz (2001)

Colloquium Mathematicae

Let ℒ be a δ-lattice in a set X, and let ν be a measure on a sub-σ-algebra of σ(ℒ). It is shown that ν extends to an ℒ-regular measure on σ(ℒ) provided ν*|ℒ is σ-smooth at ∅ and ν*(L) = inf ν*(U)|X ∖ U ∈ ℒ, Usupset L for all L ∈ ℒ. Moreover, a Choquet type representation theorem is proved for the set of all such extensions.

Extending Coarse-Grained Measures

Anna De Simone, Pavel Pták (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

In [4] it is proved that a measure on a finite coarse-grained space extends, as a signed measure, over the entire power algebra. In [7] this result is reproved and further improved. Both the articles [4] and [7] use the proof techniques of linear spaces (i.e. they use multiplication by real scalars). In this note we show that all the results cited above can be relatively easily obtained by the Horn-Tarski extension technique in a purely combinatorial manner. We also characterize the pure measures...

Extension of measures: a categorical approach

Roman Frič (2005)

Mathematica Bohemica

We present a categorical approach to the extension of probabilities, i.e. normed σ -additive measures. J. Novák showed that each bounded σ -additive measure on a ring of sets 𝔸 is sequentially continuous and pointed out the topological aspects of the extension of such measures on 𝔸 over the generated σ -ring σ ( 𝔸 ) : it is of a similar nature as the extension of bounded continuous functions on a completely regular topological space X over its Čech-Stone compactification β X (or as the extension of continuous...

Extensions of set functions.

Sergei V. Ovchinnikov, Jean Claude Falmagne (2003)

Mathware and Soft Computing

We establish a necessary and sufficient condition for a function defined on a subset of an algebra of sets to be extendable to a positive additive function on the algebra. It is algo shown that this condition is necessary and sufficient for a regular function defined on a regular subset of the Borel algebra of subsets of a given compact Hausdorff space to be extendable to a measure.

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