On continuity of invariant measures

Author:
Andrew Adler

Journal:
Proc. Amer. Math. Soc. **41** (1973), 487-491

MSC:
Primary 28A70; Secondary 43A07

MathSciNet review:
0328025

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Abstract: Main Theorem. *Let be a set of transformations on a set . The following conditions are then equivalent*:

(1) *There is a noncontinuous finitely additive measure defined on all subsets of and invariant under all transformations in* .

(2) *There is an integer such that for any finite subset of there is a finite subset of , with no more than elements, such that each in acts as a permutation on* .

**[1]**Andrew Adler and James Hamilton,*Invariant means via the ultrapower*, Math. Ann.**202**(1973), 71–76. MR**0324312****[2]**David Booth,*Ultrafilters on a countable set*, Ann. Math. Logic**2**(1970/1971), no. 1, 1–24. MR**0277371****[3]**Ching Chou,*On a conjecture of E. Granirer concerning the range of an invariant mean*, Proc. Amer. Math. Soc.**26**(1970), 105–107. MR**0260899**, 10.1090/S0002-9939-1970-0260899-X**[4]**E. E. Granirer,*On the range of an invariant mean*, Trans. Amer. Math. Soc.**125**(1966), 384–394. MR**0204551**, 10.1090/S0002-9947-1966-0204551-9**[5]**E. Granirer and Anthony T. Lau,*Invariant means on locally compact groups*, Illinois J. Math.**15**(1971), 249–257. MR**0277667****[6]**Roy C. Snell,*The range of invariant means on locally compact groups and semigroups*, Proc. Amer. Math. Soc.**37**(1973), 441–447. MR**0313481**, 10.1090/S0002-9939-1973-0313481-2**[7]**A. Sobczyk and P. C. Hammer,*A decomposition of additive set functions*, Duke Math. J.**11**(1944), 839–846. MR**0011164**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1973-0328025-9

Keywords:
Finitely additive,
invariant,
measure,
continuous,
ultrafilter

Article copyright:
© Copyright 1973
American Mathematical Society