The " weighted average " statement can be characterized thus : there is a non-negative finite Borel measure on [ 0, " ), with cumulative distribution function " g ", such that
22.
Swift attended the University of Oregon, earning a master's degree in Mathematics in 1951, before attending the University of Washington where he earned a PhD in Mathematics in 1954 under Edwin Hewitt on irregular Borel measure.
23.
Consider the Borel measure ( on the ?-algebra generated by the open sets ) in the state space of each random variable defined by assigning each Borel set the \ mathfrak P-measure of its preimage in \ mathcal F.
24.
However it also corresponds to the ( non-regular ) Borel measure that assigns measure 1 to any measurable subset of the space of ordinals less than ? that is closed and unbounded, and assigns measure 0 to other measurable subsets.
25.
This allows us to use the Riesz representation theorem and find that the dual space of ! " ( "'N "') can be identified with the space of finite Borel measures on ? "'N " '.
26.
A slightly more sophisticated formulation is as follows : suppose in addition that G is unimodular, then since \ Gamma is discrete it is also unimodular and by general theorems there exists a unique G-invariant Borel measure on G / \ Gamma up to scaling.
27.
An outer measure satisfying only the first of these two requirements is called a " Borel measure ", while an outer measure satisfying only the second requirement ( with the Borel set B replaced by a measurable set B ) is called a " regular measure ".
28.
It can also be proved that there exists a unique ( up to multiplication by a positive constant ) right-translation-invariant Borel measure \ nu satisfying the above regularity conditions and being finite on compact sets, but it need not coincide with the left-translation-invariant measure \ mu.
29.
Further generalisations are sometimes considered, for example A ^ 2 _ \ nu denotes a weighted Bergman space ( often called a Zen space ) with respect to a translation-invariant positive regular Borel measure \ nu on the closed right complex half-plane \ overline { \ mathbb { C } _ + }, that is
30.
There are many closely related variations of the theorem, as the linear functionals can be complex, real, or positive, the space they are defined on may be the unit interval or a compact space or a locally compact space, the continuous functions may be vanishing at infinity or have compact support, and the measures can be Baire measures or regular Borel measures or Radon measures or signed measures or complex measures.
