Admissible representations for probability measures

Abstract

In a recent paper, probabilistic processes are used to generate Borel probability measures on topological spaces X that are equipped with a representation in the sense of Type-2 Theory of Effectivity. This gives rise to a natural representation of the set M(X) of Borel probability measures on X. We compare this representation to a canonically constructed representation which encodes a Borel probability measure as a lower semicontinuous function from the open sets to the unit interval. This canonical representation turns out to be admissible with respect to the weak topology on M(X). Moreover, we prove that for countably based topological spaces X the representation via probabilistic processes is equivalent to the canonical representation and thus admissible with respect to the weak topology on M(X).

Are unbounded linear operators computable on the average for Gaussian measures?

“…For the case X = [0, 1], this representation has been introduced in [16]. A more general definition is studied in [12]. The results of [12,17] show that ϑ 1 M< has a number of desirable properties; we believe that it is justified to consider ϑ 1 M< as the "right" representation for Borel probability measures.…”

“…A more general definition is studied in [12]. The results of [12,17] show that ϑ 1 M< has a number of desirable properties; we believe that it is justified to consider ϑ 1 M< as the "right" representation for Borel probability measures. In the proof of Theorem 17 we shall need yet another representation of Borel probability measures: It is straightforward to derive a notation ϑ alg of the algebra A( ) generated by .…”

How constructive is constructing measures?

“…Theorem 9 (Schröder [43]) Let X be a complete computably admissible space. The map µ → δ * X µ : P(N N ) → P(X) is computable and computably invertible.…”

Computable Riesz Representation for Locally Compact Hausdorff Spaces

“…But neither such measures nor their relation to the functions of bounded variation have been studied in computable analysis. Computable Borel measures have been studied in [20,13,19,18]. But their relation to the computable measure spaces considered in [23,24,25,26] is not yet known.…”

“…There are only few publications on computable measure theory in the framework of TTE [20,13,23,19,24,19,25,26,12,18]. In the following four cases the "dual" space is studied:…”