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Formally, a metric measure space is a metric space equipped with a Borel regular measure such that every ball has positive measure. [21] For example Euclidean spaces of dimension n, and more generally n-dimensional Riemannian manifolds, naturally have the structure of a metric measure space, equipped with the Lebesgue measure.
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the σ-algebra) and the method that is used for measuring (the measure). One important example of a measure space is a probability space.
This measure space is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The σ-finite measure spaces have some very convenient properties; σ-finiteness can be compared in this respect to the Lindelöf property of topological spaces.
The term Borel space is used for different types of measurable spaces. It can refer to any measurable space, so it is a synonym for a measurable space as defined above [1] a measurable space that is Borel isomorphic to a measurable subset of the real numbers (again with the Borel -algebra) [3]
Given a Borel measure μ on a metric space X such that μ(X) > 0 and μ(B(x, r)) ≤ r s holds for some constant s > 0 and for every ball B(x, r) in X, then the Hausdorff dimension dim Haus (X) ≥ s. A partial converse is provided by the Frostman lemma: [7] Lemma: Let A be a Borel subset of R n, and let s > 0. Then the following are equivalent:
On the other hand, if X is a compact metric space, then the Wasserstein metric turns P (X) into a compact metric space. Convergence in the Radon metric implies weak convergence of measures: (,), but the converse implication is false in general.
Let < and (,,) be a measure space and consider an integrable simple function on given by = =, where are scalars, has finite measure and is the indicator function of the set , for =, …,. By construction of the integral , the vector space of integrable simple functions is dense in L p ( S , Σ , μ ) . {\displaystyle L^{p}(S,\Sigma ,\mu ).}
Given a (possibly incomplete) measure space (X, Σ, μ), there is an extension (X, Σ 0, μ 0) of this measure space that is complete. [3] The smallest such extension (i.e. the smallest σ-algebra Σ 0) is called the completion of the measure space. The completion can be constructed as follows: