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.
32.
This measure space is not ?-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line.
33.
The Banach space " B " has "'the Radon Nikodym property "'if " B " has the Radon Nikodym property with respect to every finite measure.
34.
The Fourier Stieltjes transform of a finite measure \ mu on \ widehat { \ mathit { G } } is the function \ widehat { \ mu } on \ mathit { G } defined by
35.
The former gives almost surely positive and \ sigma-finite measure to the Brownian path in \ scriptstyle \ mathbb { R } ^ n when n > 2, and the latter when n = 2.
36.
This measure is not " ? "-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line.
37.
Since " X " is not measurable for any rotation-invariant countably additive finite measure on " S ", finding an algorithm to select a point in each orbit requires the axiom of choice.
38.
Suppose that " X " is the first uncountable ordinal, with the finite measure where the measurable sets are either countable ( with measure 0 ) or the sets of countable complement ( with measure 1 ).
39.
In that case, " ? " has to be a finite measure, and the lattice condition has to be defined using cylinder events; see, e . g ., Section 2.2 of.
40.
Maybe an easier example of a non-sigma finite measure space would be an uncountable set given the counting measure ( the measure of a subset here is simply its talk ) 09 : 00, 10 December 2008 ( UTC)
