Math. Z. 196, 407-413 (1987) Mathematische Zeitschrift 9 Springer-Verlag 1987 A Decomposition of Measures in Euclidean Space Yielding Error Bounds for Quadrature Formulas Erich Novak Mathematisches Institut, Universit/it Erlangen-Ntirnberg, Bismarckstr. 1 1/2 , D-8520 Erlangen, Federal Republic of Germany 1. Introduction Let # be a finite Borel measure with a compact support supp(#)c R ~. For a >0 we define the a-size S~(#) of # to be the number S~(k t) = I]P I[' diam (supp(#)) ~, where [] # I] is the variation of # and diam(M) is the diameter of a set M in euclidean s-space. For the given measure # we want to construct a decomposition N #= ~ #1 with S,(#i)<6 i=1 of # where the supp(#i) are nonoverlapping (i.e. their interiors are disjoint). We are interested in the number N=N~(~, #) of measures which are necessary for such a decomposition. Our main result is the following. Theorem 1 s s N~(6,#)<=Cs,~.S~(#)~.g) ~+s, where cs,~ > 0 is a constant not depending on # or 0 < 6 < S~(#). Our proof of this statement uses a decomposition technique due to Calderon and Zygmund (1952). (See also de Guzmfin (1981) and Stein (1970).) Considering the usual Lebesgue measure on a cube [-a, b] s it is easy to see that the exponents in this estimate are optimal. By means of this result we can prove new error bounds for quadrature formulas. Let Ft be a finite Borel measure on E0, 1] s and let W~'s be the Sobolev class W~'~={/: [0, 1]s--*R[ Z IlD~fllp <1) I~1 =k where p k > s (imbedding condition). We show that there is a quadrature formula Q,(f)= ~ clf(xi) i=1