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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]
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms .
A sample space is usually denoted using set notation, and the possible ordered outcomes, or sample points, [5] are listed as elements in the set. It is common to refer to a sample space by the labels S, Ω, or U (for "universal set"). The elements of a sample space may be numbers, words, letters, or symbols.
In probability theory particularly in the Malliavin calculus, a Gaussian probability space is a probability space together with a Hilbert space of mean zero, real-valued Gaussian random variables. Important examples include the classical or abstract Wiener space with some suitable collection of Gaussian random variables. [1] [2]
Properties of a Gaussian probability space that do not depend on the particular choice of basis are called intrinsic and such that do depend on the choice extrensic. [1] We denote the countably infinite product of real spaces as R N = ∏ i = 1 ∞ R {\displaystyle \mathbb {R} ^{\mathbb {N} }=\prod \limits _{i=1}^{\infty }\mathbb {R} } .
Probability is the branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur. [note 1] [1] [2] A simple example is the tossing of a fair (unbiased) coin. Since the ...
The product of two standard probability spaces is a standard probability space. The same holds for the product of countably many spaces, see (Rokhlin 1952, Sect. 3.4), (Haezendonck 1973, Proposition 12), and (Itô 1984, Theorem 2.4.3). A measurable subset of a standard probability space is a standard probability space.
Σ ∗ is a filtration of the underlying probability space (Ω, Σ, ); Y is adapted to the filtration Σ ∗, i.e., for each t in the index set T, the random variable Y t is a Σ t-measurable function; for each t, Y t lies in the L p space L 1 (Ω, Σ t, ; S), i.e.