This is essentially the result of Feldheim, that any stable random vector can be characterized by a spectral measure \ Lambda ( a finite measure on \ mathbb S ) and a shift vector \ delta \ in \ mathbb R ^ d.
42.
In the following, we assume that ? is a ?-finite measure on a standard Borel " G "-space " X " such that the action of " G " respects the measure class of ?.
43.
Let ( T, \ mathcal { T }, \ mu ) be a finite measure space and suppose that, for each s \ in S, f ( s; t ) is a bounded and measurable function of t \ in T.
44.
The generalized Bochner theorem states that a measurable function on \ mathit { G } is equal, almost everywhere, to the Fourier Stieltjes transform of a non-negative finite measure on \ widehat { G } if and only if it is positive definite.
45.
Indeed, the bounded convergence theorem ( a corollary of the dominated convergence theorem ) of real analysis states that if a sequence of functions on a set of finite measure is uniformly bounded and converges pointwise, then passage of the limit under the integral is valid.
46.
You can also derive the same result using finite measures only by writing the Lebesgue measure as a limit of finite measures ( e . g . measures supported on larger and larger closed balls ) . talk ) 01 : 32, 28 February 2008 ( UTC)
47.
You can also derive the same result using finite measures only by writing the Lebesgue measure as a limit of finite measures ( e . g . measures supported on larger and larger closed balls ) . talk ) 01 : 32, 28 February 2008 ( UTC)
48.
The Poisson point process can be further generalized to what is sometimes known as the "'general Poisson point process "'or "'general Poisson process "'by using a Radon measure \ textstyle \ Lambda, which is locally-finite measure.
49.
Countably additive vector measures defined on sigma-algebras are more general than finite measures, finite signed measures, and complex measures, which are countably additive functions taking values respectively on the real interval [ 0, \ infty ), the set of real numbers, and the set of complex numbers.
50.
A measure m is s-finite if and only if there exists a sequence of finite measures ( m _ n ) _ n : m = \ sum _ { n \ in \ mathbb { N } } m _ n, that is : m is the countable sum of finite measures.
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