Aequatlones Mathematlcae 33 (1987) 1 8 - 2 2 0001-9054/87/0010184)551 50 +0 20/0 Umverslty of Waterloo © 1987 Blrkhauser Verlag, Basel Research Papers Small sets and a class of general functional equations JENS PETER REUS CHRISTENSEN AND PAL FISCHER I. Haar zero sets m an abehan Pohsh group were mtroduced in 1972 [2] The followmg result describes an important feature of these zero sets [2] THEOREM 1 (Chnstensen) Let (G, + ,d) be an abehan Pohsh group wtth tran~latton mvarlant metrtc d Let A c G be a umversally rnea~urable set whwh t~ not a Haar zero set Then {9~G A n(A + 9) ts not a Haar zero ~et} IS a netghbourhood of 0 m G Chrlstensen measurable sets and Chrlstensen measurable functions were mtroduced m 1980 [4-1 The Chnstensen measurable subsets of an abehan Pohsh group G form a a-algebra It was pomted out m [4] that this fact allows one to define Chnstensen measurabdlty of a mapping Thus, if Y is a topological space and f G ~ Y, thenfls Chrtstensen measurable lff-l(U) ts Chrtstensen measurable subset of G for every open set U of Y (We should mention that In [4] Chrlstensen measurabdtty was defined only in the case when Y is also an abehan Pohsh group, and the more general definttton can be found m [5] ) AMS (t980) subject classification Primary 43A05, 28C10 Secondary 39B70 This work was supported m part by the Damsh National Science Foundation and by the NSERC of Canada under grant A-8421 Manustrtpt retewed Januar) 20 1986 and retinal form June 20 1986