A non-probabilistic approach to Poisson spaces

1983 ◽  
Vol 93 (3-4) ◽  
pp. 181-188 ◽  
Author(s):  
Alan L. T. Paterson
Keyword(s):  
Probability Theory ◽  
Probability Measure ◽  
Harmonic Functions ◽  
Hausdorff Space ◽  
Probabilistic Approach ◽  
Compact Semigroup ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Locally Compact Semigroup ◽  
Poisson Space

SynopsisUsing techniques from probability theory, it has been established that if μ is a probability measure on a separable, locally compact group, then the space of μ-harmonic functions on the group can be identified with C(X) for some compact, Hausdorff space X. The space X is known as the Poisson space of μ. We generalise this result in the context of a measure μ on a locally compact semigroup S, in particular establishing the existence of a Poisson space for non-separable groups. The proof is non-probabilistic, and depends on properties of projections on C(K)(K compact Hausdorff). We then show that if S is compact and the support of μ generates S, then the Poisson space associated with μ, is X, where X×G×Y is the Rees product representing the kernel of S.

scholarly journals A note on the idempotent measures on countable semigroups

1970 ◽  
Vol 11 (4) ◽  
pp. 417-420
Author(s):  
Tze-Chien Sun ◽  
N. A. Tserpes
Keyword(s):  
Probability Measure ◽  
Continuous Mapping ◽  
Compact Semigroup ◽  
Probability Measures ◽  
Left Zero ◽  
Product Topology ◽  
Locally Compact ◽  
Locally Compact Semigroup ◽  
Image Position ◽  
Completely Simple

In [6] we announced the following Conjecture: Let S be a locally compact semigroup and let μ be an idempotent regular probability measure on S with support F. Then(a) F is a closed completely simple subsemigroup.(b) F is isomorphic both algebraically and topologically to a paragroup ([2], p.46) X × G × Y where X and Y are locally compact left-zero and right-zero semi-groups respectively and G is a compact group. In X × G × Y the topology is the product topology and the multiplication of any two elements is defined by , x where [y, x′] is continuous mapping from Y × X → G.(c) The induced μ on X × G × Y can be decomposed as a product measure μX × μG× μY where μX and μY are two regular probability measures on X and Y respectively and μG is the normed Haar measure on G.

Exponential Boundedness and Amenability of Open Subsemigroups of Locally Compact Groups

1994 ◽  
Vol 46 (06) ◽  
pp. 1263-1274 ◽  
Author(s):  
Wojciech Jaworski
Keyword(s):  
Probability Measure ◽  
Open Subset ◽  
Compact Support ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Complete Proof ◽  
Absolutely Continuous ◽  
Crucial Part

Abstract Let G be a connected amenable locally compact group with left Haar measure λ. In an earlier work Jenkins claimed that exponential boundedness of G is equivalent to each of the following conditions: (a) every open subsemigroup S ⊆ G is amenable; (b) given and a compact K ⊆ G with nonempty interior there exists an integer n such that (c) given a signed measure of compact support and nonnegative nonzero f ∈ L ∞(G), the condition v * f ≥ 0 implies v(G) ≥ 0. However, Jenkins‚ proof of this equivalence is not complete. We give a complete proof. The crucial part of the argument relies on the following two results: (1) an open σ-compact subsemigroup S ⊆ G is amenable if and only if there exists an absolutely continuous probability measure μ on S such that lim for every s ∈ S; (2) G is exponentially bounded if and only if for every nonempty open subset U ⊆ G.

On the Semigroup of Probability Measures of a Locally Compact Semigroup II

1987 ◽  
Vol 30 (3) ◽  
pp. 273-281 ◽  
Author(s):  
James C. S. Wong
Keyword(s):  
Convex Sets ◽  
Compact Convex ◽  
Compact Semigroup ◽  
Probability Measures ◽  
Locally Compact ◽  
Locally Compact Semigroup ◽  
Compact Semigroups ◽  
Fixed Point Properties ◽  
The Relationship ◽  
Locally Compact Semigroups

AbstractThis is a sequel to the author's paper "On the semigroup of probability measures of a locally compact semigroup." We continue to investigate the relationship between amenability of spaces of functions and functionals associated with a locally compact semigroups S and its convolution semigroup MO(S) of probability measures and fixed point properties of actions of S and MO(S) on compact convex sets.

scholarly journals Stationary countable dense random sets

2000 ◽  
Vol 32 (01) ◽  
pp. 86-100 ◽  
Author(s):  
Wilfrid S. Kendall
Keyword(s):  
Probability Theory ◽  
Polish Space ◽  
Measurable Subset ◽  
Locally Compact Group ◽  
Random Sets ◽  
Random Set ◽  
Locally Compact ◽  
Polish Spaces ◽  
Probability Zero ◽  
Group Of Symmetries

We study the probability theory of countable dense random subsets of (uncountably infinite) Polish spaces. It is shown that if such a set is stationary with respect to a transitive (locally compact) group of symmetries then any event which concerns the random set itself (rather than accidental details of its construction) must have probability zero or one. Indeed the result requires only quasi-stationarity (null-events stay null under the group action). In passing, it is noted that the property of being countable does not correspond to a measurable subset of the space of subsets of an uncountably infinite Polish space.

scholarly journals Right invariant integrals on locally compact semigroups

1964 ◽  
Vol 4 (3) ◽  
pp. 273-286 ◽  
Author(s):  
J. H. Michael
Keyword(s):  
Compact Semigroup ◽  
Compact Sets ◽  
Property A ◽  
Locally Compact ◽  
Invariant Integrals ◽  
Positive Linear Functional ◽  
Locally Compact Semigroup ◽  
Compact Semigroups ◽  
Image Position ◽  
Locally Compact Semigroups

An integral on a locally compact Hausdorff semigroup ς is a non-trivial, positive, linear functional μ on the space of continuous real-valued functions on ς with compact supports. If ς has the property: (A) for each pair of compact sets C, D of S, the set is compact; then, whenever and a ∈ S, the function fa defined by is also in . An integral μ on a locally compact semigroup S with the property (A) is said to be right invariant if for all j ∈ and all a ∈ S.

On r*-Invariant Measure on a Locally Compact Semigroup with Recurrence

1980 ◽  
Vol 23 (2) ◽  
pp. 237-239
Author(s):  
Samuel Bourne
Keyword(s):  
Invariant Measure ◽  
Borel Measure ◽  
Compact Semigroup ◽  
Locally Compact ◽  
Borel Sets ◽  
Left Group ◽  
Locally Compact Semigroup ◽  
Regular Borel Measure

A regular Borel measure μ is said to be r*-invariant on a locally compact semigroup if μ(Ba-1) = μ(B) for all Borel sets B and points a of S, where Ba-1 ={xϵS, xaϵB}. In [1] Argabright conjectured that the support of an r*-invariant measure on a locally compact semigroup is a left group, Mukherjea and Tserpes [4] proved this conjecture in the case that the measure is finite; however their method of proof fails when the measure is infinite.

POISSON SPACES FOR PROPER SEMIGROUPS OF SEMI-SIMPLE LIE GROUPS

2007 ◽  
Vol 07 (03) ◽  
pp. 273-297 ◽  
Author(s):  
JORGE N. LÓPEZ ◽  
PAULO R. C. RUFFINO ◽  
LUIZ A. B. SAN MARTIN
Keyword(s):  
Probability Measure ◽  
Lie Groups ◽  
Lie Group ◽  
Harmonic Functions ◽  
Classical Result ◽  
Continuous Functions ◽  
Empty Interior ◽  
Connected Group ◽  
Finite Center ◽  
Poisson Space

Let ν be a probability measure on a semi-simple Lie group G with finite center. Under the hypothesis that the semigroup S generated by ν has non-empty interior, we identify the Poisson space Π = G/MνAN, where bounded (l.u.c.) ν-harmonic functions in G have a one-to-one correspondence with measurable (continuous) functions in Π. This paper extends a classical result (see Furstenberg [7], Azencott [1] and others), where the semigroup generated by ν was assumed to be the whole (connected) group. We present two detailed examples.

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