By Ma J. M., Qiu R. F.

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**Additional resources for 3-Manifold Containing Separating Incompressible Surfaces of All Positive Genera**

**Sample text**

2. 1. is due to Kadison, see p. , see [1], 1,§2. 3. 2. is due to Choquet. The present proof is due to Choquet and Meyer, see [1]. The idea of the order in the first step goes back to Bishop de Leeuw and in the present form to Mokobodzki,see [1], 1,§4. , see [3], §31. 4. 2. is due to Choquet for the equivalences (a), (c) and (e) ; the others are due to Edwards, see [I], 11,§3. 5. Due to Bauer, see[l], 11,§4. 6. See [I], 1,§5. M. ALFSEN. Compact convex sets and boundary integrals. Springer-Verlag, Berlin ]971 2] N.

FINITE DIMENSION. Let X be of finite dimension. ,r n ~ ]O,1[ such that : ~" i r. x.. 11 the Dirac measure at the point y of X, x is the resultant of the discrete measure ~_ i rid(x i) which is concentrated on E(X). Does the same hold for general convexes with the word Radon probability measure instead of discrete measure? The answer is very difficult even in a as simple case as when E(X) is denumbrable. 2. THEOREM. (sketch compact convex set. e. a of open sets, and every x K X is the barycenter of a pro- on E(X).

The Choquet boundary Furthermore, is the inverse image of E(X) by the evaluation map. 2. 2. THEOREM. (a) A closed subset S of T is a ~ilov set for F if and only if S contains the Choquet boundary. (b) F admits a ~ilov boundary which is the closure of the Choquet boundary. 3. THEOREM. If T is metrizable, continuous the Choquet boundary of F is a G s set of T and, for any linear positive functional L on F there is at least one probability mea- sure e on T concentrated L(f) = @(f) on the Choquet boundary for all f ~F.

### 3-Manifold Containing Separating Incompressible Surfaces of All Positive Genera by Ma J. M., Qiu R. F.

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