For royden real analysis pdf topological space X, the collection of all Borel sets on X forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space.
Any measure defined on the Borel sets is called a Borel measure. In some contexts, Borel sets are defined to be generated by the compact sets of the topological space, rather than the open sets. In the case X is a metric space, the Borel algebra in the first sense may be described generatively as follows. The claim is that the Borel algebra is Gω1, where ω1 is the first uncountable ordinal number. To prove this claim, note that any open set in a metric space is the union of an increasing sequence of closed sets. Gm is closed under countable unions.
Note that for each Borel set B, there is some countable ordinal αB such that B can be obtained by iterating the operation over αB. However, as B varies over all Borel sets, αB will vary over all the countable ordinals, and thus the first ordinal at which all the Borel sets are obtained is ω1, the first uncountable ordinal. An important example, especially in the theory of probability, is the Borel algebra on the set of real numbers. It is the algebra on which the Borel measure is defined.
The Borel algebra on the reals is the smallest σ-algebra on R which contains all the intervals. In the construction by transfinite induction, it can be shown that, in each step, the number of sets is, at most, the power of the continuum. Let X be a topological space. B is the σ-algebra of Borel sets of X.
Mackey defined a Borel space somewhat differently, writing that it is “a set together with a distinguished σ-field of subsets called its Borel sets. However, modern usage is to call the distinguished sub-algebra measurable sets and such spaces measurable spaces. Measurable spaces form a category in which the morphisms are measurable functions between measurable spaces. Let X be a Polish space, that is, a topological space such that there is a metric d on X which defines the topology of X and which makes X a complete separable metric space. Considered as Borel spaces, the real line R, the union of R with a countable set, and Rn are isomorphic. A standard Borel space is the Borel space associated to a Polish space. A standard Borel space is characterized up to isomorphism by its cardinality, and any uncountable standard Borel space has the cardinality of the continuum.
For subsets of Polish spaces, Borel sets can be characterized as those sets which are the ranges of continuous injective maps defined on Polish spaces. Note however, that the range of a continuous noninjective map may fail to be Borel. Every probability measure on a standard Borel space turns it into a standard probability space. In contrast, an example of a non-measurable set cannot be exhibited, though its existence can be proved.
In fact, it is analytic, and complete in the class of analytic sets. Ergodic Theory and Virtual Groups”, Math. Is every sigma-algebra the Borel algebra of a topology? Tommy Norberg and Wim Vervaat, Capacities on non-Hausdorff spaces, in: Probability and Lattices, in: CWI Tract, vol.
Richard Dudley, Real Analysis and Probability. 51 “Borel sets and Baire sets”. This page was last edited on 13 March 2018, at 05:47. Nikodym theorem is a result in measure theory. The theorem is very important in extending the ideas of probability theory from probability masses and probability densities defined over real numbers to probability measures defined over arbitrary sets. It tells if and how it is possible to change from one probability measure to another.
For example, it can be used to prove the existence of conditional expectation for probability measures. The latter itself is a key concept in probability theory, as conditional probability is just a special case of it. Amongst other fields, financial mathematics uses the theorem extensively. Let ν, μ, and λ be σ-finite measures on the same measure space. Nikodym theorem makes the assumption that the measure μ with respect to which one computes the rate of change of ν is σ-finite. Consider the Borel σ-algebra on the real line. Borel set is at most a countable union of finite sets.
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