By J. Pfanzagl

The aso conception built in Chapters eight - 12 presumes that the tan gent cones are linear areas. within the current bankruptcy we gather a number of common examples the place the tangent cone fails to be a linear area. those examples are to remind the reader that an extension of the theo ry to convex tangent cones is needed. because the effects aren't wanted within the remainder of the ebook, we're extra beneficiant ab out regularity condi tions. the typical characteristic of the examples is the subsequent: Given a pre order (i.e., a reflexive and transitive order relation) on a kin of p-measures, and a subfamily i of order similar p-measures, the fa mily comprises p-measures similar with the weather of i. This frequently results in a (convex) tangent cone 1f purely p-measures greater (or smaller) than these in i are thought of, or to a tangent co ne con sisting of a convex cone and its reflexion approximately zero if either smaller and bigger p-measures are allowed. For partial orders (i.e., antisymmetric pre-orders), ireduces to a unmarried p-measure. we don't suppose the p-measures in to be pairwise similar.

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**Example text**

Several families dO, however, not share all of the additional properties discussed in Chapter 1, such as loc. aso symmetry or loc. 1. I aso convexity. 1ent spaces In this section we present examples in which the tangent space Ts(p,'ll) equals ~*(P). 1. Example. The obvious example for which Ts(p,'ll) = ~*(P) is the family which contains with P all p-measures Q equivalent to P with 6(Q;P) sufficiently small. Proof. 1. 2) We have ~(Pt,g) = 1 and Pt,g ~ 0 for all sUfficiently small t > o. Moreover, so that Pt,g E ~ if t i6 sufficiently smalI.

If it is, in addi- tion, symmetrie, then it is a linear spaee. In this ease we speak of a tangent spaae. Our general eonsiderations are mostly eoneerned with tangent spaees. An extension of the whole theory to general tangent eones is desirable. 5. Tangent spaces of induced measures Let map_, ~IJlI be a family of p-measures, T: (x,~ .... (y,[JI) a measurable and ~*T = {P*T: P E~} the family of indueed p-measures. : For any gE 9'1 (P), (pTg) oT is a eonditional expeetation of g, given T, with respeet to P.

Dij(e):= a "äS:" J (i) (',9): F i (e). k a€~. dij(e)a j o for i = 1, ••• ,q}. where In general. m o ) is of dimension k-q. Neglecting technicalities. the two modes of specifying subfamilies are equivalent: An rn-dimensional curved subfamily corresponds to a subfamily selected by q = k-m side conditions. 3. Families of symmetrie distributions Let ~ be the family of all symmetrie distributions over m whieh are equivalent to the Lebesgue measure A. t{x,P). t' (',P) skew- symmetrie about m{P). Let ~(P) denote the elass of all funetions in ~*(P) whieh are symmetrie about m{P).