On continuity of invariant measures

Author:
Andrew Adler

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

MSC:
Primary 28A70; Secondary 43A07

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

MathSciNet review:
0328025

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Abstract | References | Similar Articles | Additional Information

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]**A. Adler,*Invariant means via the ultrapower*, Math. Ann.**202**(1973), 71-76. MR**0324312 (48:2664)****[2]**D. Booth,*Ultrafilters on a countable set*, Ann. Math. Logic**2**(1970), 1-24. MR**43**#3104. MR**0277371 (43:3104)****[3]**C. Chou,*On a conjecture of E. Granirer concerning the range of an invariant mean*, Proc. Amer. Math. Soc.**26**(1970), 105-107. MR**41**#5519. MR**0260899 (41:5519)****[4]**E. Granirer,*On the range of an invariant mean*, Trans. Amer. Math. Soc.**125**(1966), 384-394. MR**34**#4390. MR**0204551 (34:4390)****[5]**E. Granirer and A. Lau,*Invariant means on locally compact groups*, Illinois J. Math.**15**(1971), 249-257. MR**43**#3400. MR**0277667 (43:3400)****[6]**R. Snell,*The range of invariant means on topological groups and semigroups*, Proc. Amer. Math. Soc. (to appear). MR**0313481 (47:2035)****[7]**A. Sobczyk and P. Hammer,*A decomposition of additive set functions*, Duke Math. J.**11**(1944), 839-846. MR**6**, 129. MR**0011164 (6:129d)**

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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