By Prof. Martin L. Silverstein (auth.)
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Example text
Infinitesimal future of T h e o r e m 4,9. 14) Proof. that there are no times implies in particlar that the sigma-algebras @x n u l l s e t s f o r i n ~I-N. x of We single out s p e c i a l c a s e of t h i s i n Gorollary to the [ k t / n < T < (k+i) t / n ] (ii). 14) follows immediately. /// for x in X-N. 15 (except that we i m i t a t e the r e c e n t l i t e r a t u r e and r e p l a c e F o l l o w i n g [35] the a d j e c t i v e " a c c e s s i b l e " by "predictable") we i n t r o d u c e 4,4, Definition.
8) G i E(pG+t [f _ pG], pG + t[f_pG] > E(pG pG ) for all real t. 8 are finite and then C a p (G I) + C a p ( G Z) - C a p ( G I UGz) - C a p ( G I N G z) G1 UGz GI =E(p , p GI which is on > 0 G I U G Z. 8) for t > 0 with G I tJ G z G1 p G1 ~Gz p ) -p playing the role of p G and f. 10. with p Gz + p theory of capacities. F o r any Borel set G I~ G 2 - p playing the role (See [3 5, chap. Z]. ) A Cap(A) = sup Cap(K) as K r u n s over the c o m p a c t s u b s e t s of A . 11. /// Proof. 11 w e can a s s u m e that A is compact.
That unless otherwise specified =F(e ) (in the transient case) are represented by their quasi-continuous versions. 6 note first that by the spectral theorem are, except possibly for quasi-continuous refinements, from Pt f LZ(dx) to F. is in For t rational and for bounded =F by construction. 5 the functions decrease to t. Next fix t > 0 everywhere. Pt and Gu bounded operators f in F the function This is also true for t real since by Ps f -~ Pt f quasi-everywhere and bounded g in F The collection of nonnegative functions as rational s such that g > 0 quasif for which Pt { m l n (f, g) } is in F__ is closed under pointwise limits and contains /~ D Ccom(__X) and t h e r e f o r e is the s e t of a l l n o n n e g a t i v e B o r e l f u n c t i o n s .