Showing papers by "Andrzej Pelc published in 1984"

The nonexistence of maximal invariant measures on abelian groups

Abstract: It is shown that every a-additive a-finite invariant measure on an abelian group has a proper a-additive invariant extension. We consider a-finite countably additive measures which vanish on points and are nonidentically zero. Throughout this paper the word "measure" will mean a measure enjoying all the above properties. A measure m defined on a a-algebra S of subsets of X is called invariant with respect to a group G of bijections of X if for any T E G and A E S the image T*(A) is an element of S and m(T*(A)) = m(A). A measure m defined on a cr-algebra of subsets of a group G is called invariant if it is invariant with respect to left translations. Sierpin'ski (quoted in Szpilrajn [7]) asked whether there exists in Euclidean ndimensional space En a maximal extension of the Lebesgue measure invariant with respect to the group of isometries of El. Hulanicki [2] proved that if (XI is less than the first real-valued measurable cardinal, IGI < IXI, and m is a measure on X invariant with respect to G and vanishing on sets of cardinality < IXj, then there exists a proper extension of m invariant with respect to G. Thus he solved Sierpinski's problem under additional set theoretic assumptions. Harazigvili [1] gave a negative answer to this question for n = 1 without any extra hypotheses. He also proved that there is no maximal measure invariant with respect to translations on any Euclidean space. In other words the group of translations of En does not carry maximal invariant measures. Our theorem is a generalisation of the above result. THEOREM. Every invariant measure on an abelian group (G, +) has a proper invariant extension. PROOF. We start with the following lemma, essentially due to Szpilrajn [7]. The easy proof is left to the reader. LEMMA. Let m be an invariant measure on G. If there exists a set E c G such that: 1. E is not a set of m measure zero. 2. For every sequence {gn: n E w} of elements of G, there exists a sequence {ho: a < w, } of elements of G such that for distinct a, /, m ([hc+ U (gn+ E) n h + U (gn + E)j) 0 t mnm w ar etnswon then the measure m has a proper invariant extensgion. Received by the editors November 1, 1982. 1980 Mathematics Subect Clasifsfiion. Primary 28C10, 43A05. Keu words and vhrases. Invariant measure. abelian group. (@)1984 American Mathematical Society 0002-9939/84 $1.00 + $.25 per page