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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]
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]
In probability theory, a measurable function on a probability space is known as a random variable. Formal definition. Let (, ...
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 ).
Every probability space gives rise to a measure which takes the value 1 on the whole space (and therefore takes all its values in the unit interval [0, 1]). Such a measure is called a probability measure or distribution. See the list of probability distributions for instances.
A measurable subset of a standard probability space is a standard probability space. It is assumed that the set is not a null set, and is endowed with the conditional measure. See (Rokhlin 1952, Sect. 2.3 (p. 14)) and (Haezendonck 1973, Proposition 5). Every probability measure on a standard Borel space turns it into a standard probability space.
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.
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: