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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]
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 ).
The empty set (considered as a measurable space) is the initial object of Meas; any singleton measurable space is a terminal object. There are thus no zero objects in Meas. The product in Meas is given by the product sigma-algebra on the Cartesian product. The coproduct is given by the disjoint union of measurable spaces.
For (,) a measurable space, a sequence μ n is said to converge setwise to a limit μ if = ()for every set .. Typical arrow notations are and .. For example, as a consequence of the Riemann–Lebesgue lemma, the sequence μ n of measures on the interval [−1, 1] given by μ n (dx) = (1 + sin(nx))dx converges setwise to Lebesgue measure, but it does not converge in total variation.
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:
In any measure space (,) with a non-measurable set,, one can construct a non-measurable indicator function: : (,), = {, where is equipped with the usual Borel algebra. This is a non-measurable function since the preimage of the measurable set { 1 } {\displaystyle \{1\}} is the non-measurable A . {\displaystyle A.}
In probability theory, a probability space or a probability triple (,,) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a die. A probability space consists of three elements: [1] [2]
The space of all countable ordinals with the topology generated by "open intervals" is a locally compact Hausdorff space. The measure ("Dieudonné measure") that assigns measure 1 to Borel sets containing an unbounded closed subset and assigns 0 to other Borel sets is a Borel probability measure whose support is empty.