The main purpose of this article is to study the symmetric martingale property and capacity defined by G-expectation introduced by Peng (cf. http://arxiv.org/PS_cache/math/pdf/0601/0601035v2.pdf ) in 2006. We show that the G-capacity can not be dynamic, and also demonstrate the relationship between symmetric G-martingale and the martingale under linear expectation. Based on these results and path-wise analysis, we obtain the martingale characterization theorem for G Brownian motion without Markovian assumption. This theorem covers the Levy's martingale characterization theorem for Brownian motion, and it also gives a different method to prove Levy's theorem.

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Martingale Property and Capacity under G-Framework

The regularity of borel measures

Conditions for the uniform continuity of a family of weakly regular set functions defined on an algebra Σ ⊃ η of subsets of a σ-topological space (T,η) and taking values in an arbitrary topological space are found

Uniform Continuity of a Family of Weakly Regular Set Functions on Topological Spaces

Let S be a locally compact semigroup and T a locally compact subsemigroup of S. It is shown that, for a large class of semigroups S, if T is 'reasonably large', the absolutely continuous measures on T extend to measures whose translates by some points of S are absolutely continuous on S. As a consequence we prove the absolute continuity of translates of some measures related to a locally quasi-invariant measure.

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Absolutely continuous and locally quasi-invariant measures on locally compact semigroups

Let X be a regular topological space, K a collection of bounded regular measures defined on the Borel sets of X. The following conditions are equivalent. (1) Let M(X) denote the Borel measures, M(X) the nonnegative members of M(X). There is a A EM(X) + such that K is uniformly X-continuous. (2) If IUnIn = 1, 2,. .. I is a disjoint sequence of open sets, then limn-4Un) = 0 uniformly for ,u E K. (3) If E is a Borel subset of X and E > 0, there is a compact set F C E such that 1L|(E F) 0 we can find a compact set K C E such that Iy(E-K)I 0, there is a 8 > 0 such that if E is a Borel set with A(E) < 8, then tL(E)I < E for

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James D. Stein

Papers

Uniform absolute continuity in spaces of set functions