For the duality between ?-finite measure spaces and commutative von Neumann algebras, noncommutative von Neumann algebras are called " non-commutative measure spaces ".
22.
For some non-negative finite measure ? on the circle : in this case, if non-zero, ? must be singular with respect to Lebesgue measure.
23.
More generally the transition rates are given in form of a finite measure c _ \ Lambda ( \ eta, d \ xi ) on S ^ \ Lambda.
24.
The theorem states that if a uniformly bounded sequence of functions converges pointwise, then their integrals on a set of finite measure converge to the integral of the limit function.
25.
This condition is also necessary if \ mathcal { T } contains all singleton sets and F is required to be a single crossing function for any finite measure \ mu.
26.
Analogously, a set in a measure space is said to have a " ?-finite measure " if it is a countable union of sets with finite measure.
27.
Analogously, a set in a measure space is said to have a " ?-finite measure " if it is a countable union of sets with finite measure.
28.
An abstract Wiener space is a mathematical object in measure theory, used to construct a " decent ", strictly positive and locally finite measure on an infinite-dimensional vector space.
29.
A Borel measure ? on X is boundedly finite if ? ( A ) M _ X be the space of all boundedly finite measures on \ mathfrak { B } ( X ).
30.
Nonzero finite measures are analogous to probability measures in the sense that any finite measure is proportional to the probability measure \ frac { 1 } { \ mu ( X ) } \ mu.
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