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U (An n C 1) and L = (A In L) u ... (Ai\C I ) ~ L/lexlAi n L) i= I = A(L). i=l On the other hand, given e > 0, there exist compact subsets K i £; Ai\C I such that /lexlAi\CI) - /lexlKi) < eln, and K = I K i is a decomposition of the compact set K £; C 2 \ C I ; hence Ui= n A(K) = n L/lexlKi) > L/lexlAi\CI) - e i=l = A(C 2 ) i=l - A(C I ) - e. Let /l be the unique extension of A to a Radon measure on X. Then if B is a Borel subset of some Gex , we have /l(C) = /lex(C) for each compact subset C £; B and therefore /l(B) = /lex(B).

Definition. Let X be a compact subset of E and let J1 E M~(X) be a Radon probability measure on X. A point bEE is called the barycentre of J1 if and only if f(b) = Ix f dJ1 for all j' E E'. Remark. There exists at most one barycentre of J1 E M~(X). f E E', and since E' separates the points of E we find b l = b2 . f gravity and resultant. The barycentre is "the value of" the vector integral Ix x dJ1(x), but such a vector integral need not always converge in E, so a barycentre need not exist. 3. Proposition.

There exists at most one barycentre of J1 E M~(X). f E E', and since E' separates the points of E we find b l = b2 . f gravity and resultant. The barycentre is "the value of" the vector integral Ix x dJ1(x), but such a vector integral need not always converge in E, so a barycentre need not exist. 3. Proposition. Let X be a compact subset of E such that K = conv(X) is compact. for every J1 E M~(X) the barycentre exists and belongs to K. Conversely, every point x E K is barycentre of some J1 E M ~ (X).