If a compact convex set K has an inner part A then there is a selection of pairwise boundedly absolutely continuous representing measures for A if and only if K is finite dimensional. Let K denote a compact convex set in a LCTVS, A(K) the affine continuous real functions on K, ?)(K) the set of regular Borel probability measures on K. Let (P: Y(K)--K be the map which associates to each measure ,t its barycentre. Then (D is affine, weak* continuous, and onto K. If (P(,u)=x we say ,u represents x. If L is any convex set, x, y e L and r>O, we say [x, y] ewtends by r in L if x + r(x-y) e L and j + r(y-x) e L. We write x,--y if 3 r > O such that [x, Y] extends by r in L. This is an equivalence relation on L and the equivalence classes are the parts of L. It is easy to show that (D carries parts into parts: If rI is a part of ?@(K) then (P(H) is contained in a part of K. Conversely if A is a part of K and F is any finite subset of A then there exists a part 17 of bY(K) such that Fc ()(P(). Indeed if F= fx2, . , x I choose yj and z, in K such that xl e (ye, z), the open line segment with endpoints Ji and z, and xl e (y, x1) (2?i<n). If (Du )=y= and (D(v?)=z, for ,u v, e ?9(K), then the part HI containing I (u?+v)/(2n-2) satisfies Fce((HI). Indeed since x1 e (.Yi, z) for each i, we can clearly find a representing measure () for xl in H. Since xl e (y, xj1), an affine combination of pi and a) yields a representing measure for xl in fl. Thus if A is a part of K one might ask whether (1) A = (P(H) for some part II of 9a(K). Indeed Bear posed this question in [3] and reproduced an example of Har'kova [4] to show that (1) need not hold if ?Y(K) is replaced by Y(F) where F is the Shilov boundary of A(K). Since two probability measures ,t and v on K are in the same part of ??(K) if and only if ,ukv and v_k,u for some k, condition (1) asserts Received by the editors September 11, 1970. AMS (MOS) subject classJilcations (1970). Primary 52A20, 31 BI1; Secondary 46E10, 28A40. ?c American Mathematical Society 1973