Two Kinds of Invariance of Full Conditional Probabilities

Alexander R. Pruss. "Two Kinds of Invariance of Full Conditional Probabilities." Bulletin of the Polish Academy of Sciences. Mathematics 61.3 (2013): 277-283. <http://eudml.org/doc/281213>.

@article{AlexanderR2013,
abstract = {Let G be a group acting on Ω and ℱ a G-invariant algebra of subsets of Ω. A full conditional probability on ℱ is a function P: ℱ × (ℱ∖\{∅\}) → [0,1] satisfying the obvious axioms (with only finite additivity). It is weakly G-invariant provided that P(gA|gB) = P(A|B) for all g ∈ G and A,B ∈ ℱ, and strongly G-invariant provided that P(gA|B) = P(A|B) whenever g ∈ G and A ∪ gA ⊆ B. Armstrong (1989) claimed that weak and strong invariance are equivalent, but we shall show that this is false and that weak G-invariance implies strong G-invariance for every Ω, ℱ and P as above if and only if G has no non-trivial left-orderable quotient. In particular, G = ℤ provides a counterexample to Armstrong's claim.},
author = {Alexander R. Pruss},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {full conditional probability; -invariance; left-orderable group; probability exchange rate},
language = {eng},
number = {3},
pages = {277-283},
title = {Two Kinds of Invariance of Full Conditional Probabilities},
url = {http://eudml.org/doc/281213},
volume = {61},
year = {2013},
}

TY - JOUR
AU - Alexander R. Pruss
TI - Two Kinds of Invariance of Full Conditional Probabilities
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2013
VL - 61
IS - 3
SP - 277
EP - 283
AB - Let G be a group acting on Ω and ℱ a G-invariant algebra of subsets of Ω. A full conditional probability on ℱ is a function P: ℱ × (ℱ∖{∅}) → [0,1] satisfying the obvious axioms (with only finite additivity). It is weakly G-invariant provided that P(gA|gB) = P(A|B) for all g ∈ G and A,B ∈ ℱ, and strongly G-invariant provided that P(gA|B) = P(A|B) whenever g ∈ G and A ∪ gA ⊆ B. Armstrong (1989) claimed that weak and strong invariance are equivalent, but we shall show that this is false and that weak G-invariance implies strong G-invariance for every Ω, ℱ and P as above if and only if G has no non-trivial left-orderable quotient. In particular, G = ℤ provides a counterexample to Armstrong's claim.
LA - eng
KW - full conditional probability; -invariance; left-orderable group; probability exchange rate
UR - http://eudml.org/doc/281213
ER -