ENTRY MEASURE THEORY Authors: Oliver Knill: 2003 Literature: measure theory +------------------------------------------------------------ | analytic set +------------------------------------------------------------ An analytic set in a complete seperable metric space is the continuous image of a Borel set. Also called A-set. Any A-set is Lebesgue measurable. Any uncountable A-set topologically contains a perfect Cantor set. Suslins criterium tells that an analytic set is a Borel set if and only if its complement is also an analytic set. +------------------------------------------------------------ | atom +------------------------------------------------------------ An atom is a measurable set Y of positive measure in a measure space such that every subset Z of Y has either zero or the same measure. Often an atom consists of only one point. More generally, an atom is minimal, non-zero element in a Boolean algebra. +------------------------------------------------------------ | atom +------------------------------------------------------------ A property which holds up to a set of measure zero is said to hold almost everywhere (= almost surely). +------------------------------------------------------------ | Banach-Tarski theorem +------------------------------------------------------------ The Banach-Tarski theorem: a ball in Euclidean space of dimension 3 can be decomposed into finitely many sets and rearranged by rigid motion to obtain two balls. The +------------------------------------------------------------ | barycentre +------------------------------------------------------------ The barycentre of a Lebesgue measurable set S in an Euclidean space is the point int_S x dx. +------------------------------------------------------------ | Boolean algebra +------------------------------------------------------------ A Boolean algebra is a set S with two binary operations + and * which are commutative monoids (S,+,0), (S,*,1) and satisfy the two distributive laws (x*(y+z)=x*y + y*z, x+(y*z) =(x+y)*(x+z) as well as the complementary laws x*x=1, y+y=0. A Boolean algebra is especially an algebra. Examples are the algebra of classes, where + is the union and * is the intersection or the algebra of propositions, for which + is and and * is or. +------------------------------------------------------------ | Boolean ring +------------------------------------------------------------ A Boolean ring is a ring in which every member is idempotent. +------------------------------------------------------------ | Borel-Cantelli lemma +------------------------------------------------------------ The Borel-Cantelli lemma: if Y_n are events in a probability space and the sum of their probabilities is finite, then the probability that infinitely many events occur is zero. If the events are independent and the sum of their probabilities is infinite, then the probability that infinitely many events occur is one. +------------------------------------------------------------ | Borel measure +------------------------------------------------------------ A Borel measure is a measure on the sigma-algebra of Borel sets. +------------------------------------------------------------ | Borel set +------------------------------------------------------------ A Borel set (=Borel measurable set) in a topological space is an element in the smallest sigma-algebra which contains all compact sets. Borel sets are also called B-sets. One can say that a B-set is a set which can be obtained of not more than a countable number of operations of union and intersection of closed open sets in a topological space. Borel sets are special cases of analytic sets. +------------------------------------------------------------ | Borel set +------------------------------------------------------------ The smallest sigma-algebra A of subsets of a topological space (X,O) containing O is called a Borel sigma-algebra. +------------------------------------------------------------ | absolutely continuous +------------------------------------------------------------ A measure mu is absolutely continuous to a measure nu if nu(Y)=0 implies mu(Y)=0. +------------------------------------------------------------ | centre of mass +------------------------------------------------------------ The centre of mass (=barycentre) of a Borel measure mu in a Euclidean space X is the point overlinex= int_X x mu(x). For example, if mu is supported on finitely many points x_i and m_i = mu(x_i) then overlinex = sum_i m_i x_i. If mu is the mass distribution of a body, then its centre of mass is called the centre of gravity. +------------------------------------------------------------ | abstract integral +------------------------------------------------------------ abstract integral. Denote by L,L^+ the set of measureable maps from a measure space (X,A,mu) to the real line (R,B), where B is the Borel sigma-algebra on R,R^+. For f in S= f=sum_i=1^n alpha_i . 1_A_i alpha_i in R , define int_X f dmu := sum_a in f(X) a . muX=a. For f in L^+ define int_X f dmu = sup_g in S int_X g dmu . For f in L finally define int f = int f^+ - int f^-, where f^+(x)=max(f(x),0) and f^-(x)=-(-f)^+(x). +------------------------------------------------------------ | abstract integral +------------------------------------------------------------ A sigma-additive function mu: A arrow 0,infinity on a measurable space (X,A) is called a measure. It is called a finite measure if mu(X)