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.
The theory of standard probability spaces was started by von Neumann in 1932[1] and shaped by Vladimir Rokhlin in 1940.
[2] For modernized presentations see (Haezendonck 1973), (de la Rue 1993), (Itô 1984, Sect.
Nowadays standard probability spaces may be (and often are) treated in the framework of descriptive set theory, via standard Borel spaces, see for example (Kechris 1995, Sect.
Standard probability spaces are used routinely in ergodic theory.
are isomorphic (being endowed naturally with sigma-fields and probability measures).
to an interval with Lebesgue measure, a finite or countable set of atoms, or a combination (disjoint union) of both.
2.4 (p. 20)), (Haezendonck 1973, Proposition 6 (p. 249) and Remark 2 (p. 250)), and (de la Rue 1993, Theorem 4-3).
In (Petersen 1983, Definition 4.5 on page 16) the measure is assumed finite, not necessarily probabilistic.
However, the integral of a white noise function from 0 to 1 should be a random variable distributed N(0, 1).
) are in a natural one-to-one correspondence with events and random variables on the probability space
be injective and generating, then the following two conditions are equivalent: See also (Itô 1984, Sect.
The four cases treated above are mutually equivalent, and can be united, since the measurable spaces
we get the property well known as being countably generated (mod 0), see (Durrett 1996, Exer.
2.5), (Haezendonck 1973, Corollary 2 on page 253), (de la Rue 1993, Theorems 3-4 and 3-5).
This property does not hold for the non-standard probability space dealt with in the subsection "A superfluous measurable set" above.
A standard probability space may contain a null set of any cardinality, thus, it need not be countably separated.
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
(of sequences; endowed with the product topology) into a standard probability space.
The same holds for the product of countably many spaces, see (Rokhlin 1952, Sect.
As a result, a number of well-known facts have special 'conditional' counterparts.
3.1), which is basically the same as conditional probability measures (see Itô 1984, Sect.
3.5), disintegration of measure (see Kechris 1995, Exercise (17.35)), and regular conditional probabilities (see Durrett 1996, Sect.
Equivalence classes of measurable subsets of a probability space form a normed complete Boolean algebra called the measure algebra (or metric structure).
"Standard probability space", Encyclopedia of Mathematics, EMS Press, 2001 [1994]