Standard probability space
In probability theory, a standard probability space, also called Lebesgue–Rokhlin probability space or just Lebesgue space (the latter term is ambiguous) is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin in 1940. Informally, it is a probability space consisting of an interval and/or a finite or countable number of atoms. The theory of standard probability spaces was started by von Neumann in 1932 and shaped by Vladimir Rokhlin in 1940. Rokhlin showed that the unit interval endowed with the Lebesgue measure has important advantages over general probability spaces, yet can be effectively substituted for  many of these in probability theory. The dimension of the unit interval is not an obstacle, as was clear already to Norbert Wiener. He constructed the Wiener process (also called Brownian motion) in the form of a measurable map from the unit interval to the space of continuous functions. Short history. The theory of standard probability spaces was started by von Neumann in 1932 and shaped by Vladimir Rokhlin in 1940. For modernized presentations see , , and . Nowadays standard probability spaces may be (and often are) treated in the framework of descriptive set theory, via standard Borel spaces, see for example  . This approach is based on the isomorphism theorem for standard Borel spaces . An alternate approach of Rokhlin, based on measure theory, neglects null sets, in contrast to descriptive set theory. Standard probability spaces are used routinely in ergodic theory, Definition. One of several well-known equivalent definitions of the standardness is given below, after some preparations. All probability spaces are assumed to be complete. Isomorphism. An isomorphism between two probability spaces formula_1, formula_2 is an invertible map formula_3 such that formula_4 and formula_5 both are (measurable and) measure preserving maps. Two probability spaces are isomorphic if there exists an isomorphism  between them. Isomorphism modulo zero. Two probability spaces formula_1, formula_2 are isomorphic formula_8 if there exist null sets formula_9, formula_10 such that the probability spaces formula_11, formula_12 are isomorphic (being endowed naturally with sigma-fields and probability measures). Standard probability space. A probability space is standard, if it is isomorphic formula_8 to an interval with Lebesgue measure, a finite or countable set of atoms, or a combination (disjoint union) of both. See , , and . See also , and . In the measure is assumed finite, not necessarily probabilistic. In atoms are not allowed. Examples of non-standard probability spaces. A naive white noise . The space of all functions formula_14 may be thought of as the product formula_15 of a continuum of copies of the real line formula_16. One may endow formula_16 with a probability measure, say, the standard normal distribution formula_18, and treat the space of functions as the product formula_19 of a continuum of identical probability spaces formula_20. The product measure formula_21 is a probability measure on formula_15. Many non-experts are inclined to believe that formula_21 describes the so-called white noise. However, it does not. For the white noise, its integral from 0 to 1 should be a random variable  distributed "N"(0, 1). In contrast, the integral (from 0 to 1) of formula_24 is undefined. Even worse, "ƒ" fails to be almost surely measurable. Still worse, the probability of "ƒ" being measurable is undefined. And the worst thing: if "X" is a random variable distributed (say) uniformly on (0, 1) and independent of "ƒ", then "ƒ"("X") is not a random variable at all! (It lacks measurability.) A perforated interval. Let formula_25 be a set whose inner Lebesgue measure is equal to 0, but outer Lebesgue measure is equal to 1 (thus, formula_26 is nonmeasurable to extreme). There exists a probability  measure formula_27 on formula_26 such that formula_29 for every Lebesgue measurable formula_30. (Here formula_31 is the Lebesgue measure.) Events and random variables on the probability space formula_32 (treated formula_8) are in a natural one-to-one correspondence with events and random variables on the probability space formula_34. Many non-experts are inclined to conclude that the probability space formula_32 is as good as formula_34. However, it is not. A random variable formula_37 defined by formula_38 is distributed uniformly on formula_39. The conditional measure, given formula_40, is just a single atom (at formula_41), provided that formula_34 is the underlying probability space . However, if formula_32 is used instead, then the conditional measure does not exist when formula_44. A perforated circle is constructed similarly. Its events and random variables are the same as on the usual circle. The group of rotations acts on them naturally. However, it fails to act on the perforated circle. See also . A superfluous measurable set. Let formula_25 be as in the previous example. Sets of the form formula_46 where formula_47 and formula_48 are arbitrary Lebesgue measurable sets, are a σ-algebra formula_49 it contains the Lebesgue σ-algebra and formula_50 The formula gives the general form of a  probability measure formula_27 on formula_53 that extends the Lebesgue measure; here formula_54 is a parameter. To be specific, we choose formula_55 Many non-experts are inclined to believe that such an extension of the Lebesgue measure is at least harmless. However, it is the perforated interval in disguise. The map is an isomorphism between formula_57 and the perforated interval corresponding to the set another set of inner Lebesgue measure 0 but outer Lebesgue measure 1. See also . A criterion of standardness. Standardness of a given probability space formula_59 is equivalent to a certain property of a measurable map formula_4 from  formula_59 to a measurable space formula_62 The answer (standard, or not) does not depend on the choice of formula_63 and formula_4. This fact is quite useful; one may adapt the choice of formula_63 and formula_4 to the given formula_67 No need to examine all cases. It may be convenient to examine a random variable formula_68 a random vector formula_69 a random sequence formula_70 or a sequence of events formula_71 treated as a sequence of two-valued random variables, formula_72 Two conditions will be imposed on formula_4 (to be injective, and generating). Below it is assumed that such formula_4 is given . The question of its existence will be addressed afterwards. The probability space formula_59 is assumed to be complete (otherwise it cannot be standard). A single random variable. A measurable function formula_76 induces a pushforward measure formula_77, – the probability measure formula_78 on formula_79 defined by i.e. the distribution of the random variable formula_82. The image formula_83 is always a set of full outer measure, but its inner measure can differ (see "a perforated interval"). In other words, formula_83 need not be a set of full measure formula_86 A measurable function formula_76 is called "generating" if formula_88 is the completion with  respect to formula_89 of the σ-algebra of inverse images formula_90 where formula_91 runs over all Borel sets. "Caution."   The following condition is not sufficient for formula_4 to be generating: for every formula_93 there exists a Borel set formula_91 such that formula_95 (formula_96 means symmetric difference). Theorem. Let a measurable function formula_76 be injective and generating, then the following two conditions are equivalent: See also . A random vector. The same theorem holds for any formula_101 (in place of formula_102). A measurable function formula_103 may be thought of as a finite sequence of random variables formula_104 and formula_105 is generating if  and only if formula_106 is the completion of the σ-algebra generated by formula_107 A random sequence. The theorem still holds for the space formula_108 of infinite sequences. A measurable function formula_109 may be thought of as an infinite sequence of random variables formula_110 and formula_105 is generating if and only if formula_106 is the completion of the σ-algebra generated by formula_113 A sequence of events. In particular, if the random variables formula_114 take on only two values 0 and 1, we deal with a measurable function formula_115 and a sequence of sets formula_116 The function formula_105 is generating  if and only if formula_106 is the completion of the σ-algebra generated by formula_119 In the pioneering work sequences formula_120 that correspond to injective, generating formula_105 are called "bases" of the probability space formula_100 (see ). A basis is called complete mod 0, if formula_123 is of full measure formula_124 see . In the same section Rokhlin proved that if a probability space is complete mod 0 with respect to some basis, then it is complete mod 0 with respect to every other basis, and defines "Lebesgue spaces" by this completeness property. See also and . Additional remarks. The four cases treated  above are mutually equivalent, and can be united, since the measurable spaces formula_125 formula_126 formula_108 and formula_128 are mutually isomorphic; they all are standard measurable spaces (in other words, standard Borel spaces). Existence of an injective measurable function from formula_59 to a standard measurable space formula_63 does not depend on the choice of formula_131 Taking formula_132 we get the property well known as being "countably separated" (but called "separable" in ). Existence of a generating measurable function from formula_59 to a standard measurable space formula_63 also does not depend on the choice of formula_131 Taking formula_132 we get the property well  known as being "countably generated" (mod 0), see . Every injective measurable function from a "standard" probability space to a "standard" measurable space is generating. See , , . This property does not hold for the non-standard probability space dealt with in the subsection "A superfluous measurable set" above. "Caution."   The property of being countably generated is invariant under mod 0 isomorphisms, but the property of being countably separated is not. In fact, a standard probability space formula_59 is countably separated if and only if the cardinality of formula_138 does not exceed continuum (see ). A standard probability space may contain a null set  of any cardinality, thus, it need not be countably separated. However, it always contains a countably separated subset of full measure. Equivalent definitions. Let formula_59 be a complete probability space such that the cardinality of formula_138 does not exceed continuum (the general case is reduced to this special case, see the caution above). Via absolute measurability. Definition.   formula_59 is standard if it is countably separated, countably generated, and absolutely measurable. See and . "Absolutely measurable" means: measurable in every countably separated, countably generated probability space containing it. Via perfectness. Definition.   formula_59 is standard if it is countably separated and perfect. See  . "Perfect" means that for every measurable function from formula_59 to formula_102 the image measure is regular. (Here the image measure is defined on all sets whose inverse images belong to formula_88, irrespective of the Borel structure of formula_102). Via topology. Definition.   formula_59 is standard if there exists a topology formula_148 on formula_138 such that See . Verifying the standardness. Every probability distribution on the space formula_157 turns it into a standard probability space. (Here, a probability distribution means a probability measure defined initially on the Borel sigma-algebra and completed.) The same holds on every Polish space, see , , , and . For example , the Wiener measure turns the Polish space formula_158 (of all continuous functions formula_159 endowed with the topology of local uniform convergence) into a standard probability space. Another example: for every sequence of random variables, their joint distribution turns the Polish space formula_160 (of sequences; endowed with the product topology) into a standard probability space. The product of two standard probability spaces is a standard probability space. The same holds for the product of countably many spaces, see , , and . 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 and . Every probability measure on a standard Borel space turns it into a standard probability space. Using the standardness. Regular conditional probabilities. In the discrete setup, the conditional probability is another probability measure, and the conditional expectation may be treated as the (usual) expectation with respect to the conditional measure, see conditional expectation. In the non-discrete setup, conditioning is often treated indirectly, since the condition may have probability 0, see conditional expectation. As a result, a number of well-known facts have special 'conditional' counterparts. For example: linearity of  the expectation; Jensen's inequality (see conditional expectation); Hölder's inequality; the monotone convergence theorem, etc. Given a random variable formula_161 on a probability space formula_59, it is natural to try constructing a conditional measure formula_163, that is, the conditional distribution of formula_164 given formula_165. In general this is impossible (see ). However, for a "standard" probability space formula_59 this is possible, and well known as "canonical system of measures" (see ), which is basically the same as "conditional probability measures" (see ), "disintegration of measure" (see ), and "regular conditional probabilities" (see ). The conditional Jensen's inequality is just the (usual) Jensen's  inequality applied to the conditional measure. The same holds for many other facts. Measure preserving transformations. Given two probability spaces formula_1, formula_2 and a measure preserving map formula_3, the image formula_170 need not cover the whole formula_171, it may miss a null set. It may seem that formula_172 has to be equal to 1, but it is not so. The outer measure of formula_170 is equal to 1, but the inner measure may differ. However, if the probability spaces formula_1, formula_2 are "standard " then formula_176, see . If formula_4 is also one-to-one then every formula_178 satisfies formula_179, formula_180. Therefore , formula_5 is measurable (and measure preserving). See and . See also . "There is a coherent way to ignore the sets of measure 0 in a measure space" . Striving to get rid of null sets, mathematicians often use equivalence classes of measurable sets or functions. Equivalence classes of measurable subsets of a probability space form a normed complete Boolean algebra called the "measure algebra" (or metric structure). Every measure preserving map formula_3 leads to a homomorphism formula_183 of measure algebras; basically, formula_184 for formula_185. It may seem that every homomorphism of measure algebras has to correspond to some measure preserving map, but  it is not so. However, for "standard" probability spaces each formula_183 corresponds to some formula_4. See , , .