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Extra info for A Dirichlet Problem for Distributions and Specifications for Random Fields

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3 we may assume that A and every A , n € IN , is finely closed. Let G := fl o (A ) . N M ' P V n Clearly it suffices to show that G <= a M (A) . M,P y For + y € M„ E Let k € IN , yj,... f y k € M £ , and let X denote a 8-measurable version of u Z := X u 2 € L (PTT) . V k n X i-1 y i We intend to show that E (Z|G) is a (A)-measurable. By the reverse martingale convergence theorem it follows that E Let n € IN . ,. , J. can be written as a sum of terms, each having the following form k a. k 3.

XuAn ) X n (X An ) L , a X y y i-1 i i-1 i yi n 3 x € IN U {0} . 1 that P.. u. - y. n l V Mi v M l A i Since all involved 8-measurable functions on &. k follows that n (X A,,) . s. on V% V 1*1 k 1=1 a. i»l l X A y£n i vc E are Gaussian distributed, it o M „ (A) . M,P V i * V 3. 1. M,Pyv n' * (X An) c™ converges to X A Mi n \ii 1 < i < k . 12 ] that the sequence

E. (iv) iyO Let bourhood C € fi(U) and V of P(V) . Then 3U with n € fl(U) and = C n € P1 V (v) with UC . be such that there exists an open neigh- . i on = (p € P(U U V) . e. = J 1^(5^) cp dX d for every such that (p 6 P(U U V) . £ € Pf V of 3U ,. (U)) . ' be such that £ € fl(U) and (vii) Let Then U c V £t is representd by a harmonic function. ^(C) - € . and let ft(V) and ^(flOO) <= fl(u) and lL be constructed as above for H ^ H y U ) ) - HyCO for every Proof. 5 (ii) we conclude that for U .

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