By A. A. Milyutin and N. P. Osmolovskii

The speculation of a Pontryagin minimal is constructed for difficulties within the calculus of diversifications. the appliance of the thought of a Pontryagin minimal to the calculus of diversifications is a particular characteristic of this publication. a brand new concept of quadratic stipulations for a Pontryagin minimal, which covers damaged extremals, is built, and corresponding adequate stipulations for a powerful minimal are acquired. a few classical theorems of the calculus of diversifications are generalized

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There are three essential ingredients in the proofs of such results. The first involves delicate, transfinite induction arguments, which are used to construct certain special subsets of the group that, in a sense, generate "many" left invariant means. ) probability measures on a compact space. 11). Each mean in,. can be regarded as "supported" on the set A,. ) = I or, alternatively, that m,-(i0) = m4XA,) (0 E Since A, n A. = 0 if r # a, the m,'s are "disjointly supported" and are therefore different.

One readily checks that x(f * n) _ (x * f)*nforallxEG. 9) DEFINITION. (G), f e P(G)). The set of topologically left invariant means is denoted by £t (G), and is clearly a weak* compact, convex subset of Wt(G). Further, if m E £t(G) and x E G, then xm = x * (f * m) = (x * f) * m = m since x * f E P(G). So £t(G) C £(G). It is readily checked that if G is discrete, then every function in P(G) is a sum of point masses so that £t(G) = £(G). However as we shall see in Chapter 7, the two sets are very different in general for nondiscrete locally compact groups.

C = 0. Further, G has a natural left action on ;BG, and m E £(G) a rfi. is a probability measure that is invariant under the action of G. The whole philosophy is simple: we shift from studying a bad (finitely additive) measure on a good set (G) to studying a good (that is, countably additive) measure on a complicated space ;6G. ) 12 INTRODUCTION Reverting to the m,'s and A, 's above, we readily see that the fact that m,(A,) = 1 implies that the support of m, is contained in the closure A; of A, in 6G.

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