By Palais, Richard

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**Extra resources for A Global Formulation Of Lie Theory of Transformational Groups**

**Example text**

The above definitions of course make sense for an arbitrary real vector space without topology. 1' ••. , An be numbers E [0, 1] with with 1 Ai = 1. e. 1 = L AiGxi' i= 1 56 2. 2. 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.

Y), and for the functions g' := t(g 1\ i), h' := t(h 1\ i) we see that g'l U = 1 = h'l V whereas g'l V < and also h'l U < t. Now it is clear that t f:= [g' - (h' - g')+J+ is one on U and zero on V. c such thatfx,yl U x = O,fx,yl ~ = 1. Let us first fix a point y E L, then there is a finite subset {Xl' ... ,xn } ~ K such that K ~ U Xl U ... U U x n • The function fy:= min{fxI,y, ... c, equals one in some neighbourhood ~ of y and is zero on all of K. c with l KuL ~ g; then hy := (g 1\ 1 - fy) + is zero on ~ and one on K.

1(G\K) 8. e. we have the desired equality. If A E 81(X), B E 81(Y) and C is a compact subset of Ax B, then the projections K:= nx(C) and L:= ny(C) are still compact and C £ K x L £ A x B, implying K(A x B) = sup{K(C)1 C £ A x B, C E ff(Z)} = sup{K(K x L)IK £ A, L £ B, K E ff(X), L = sup{*(K, L)IK £ A, L £ B,KE ff(X), L E E ff(Y)} ff(Y)} = (A, B), using in the last equality once more that is a Radon bimeasure. We also see from the preceding argument that K is indeed uniquely determined from its values on products of compact sets (still assuming (X, Y) < (0). *