By D. Williams
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Extra info for Stochastic integrals (Proc. Durham 1980)
E(f, g) depends only on the equivalence classes f and Abusing notation, we set e(7, 9) = e(f, g), where f and g are chosen in the classes of f and 9. 5 Proposition. Suppose that (X, A, p) is a measure space and Y is a metric space. 2. Set e(f, 9) 4 (f,9) = 1+e(f,9) Then d,, is a distance on M,,. PROOF. 3 shows that e satisfies the axioms for a distance, except that e may assume the value +oo. We use a construction common in topology; let k(t) 1 + t' t E R+, k(+oo) = I. It is elementary to verify that the function t i-+ k(t) satisfies k(t1 + t2) < k(ti) + k(t2), t1, t2 > 0.
To f , and hence IITn(hq) - Tn(f)IILI 0. It follows that. 1 Theorem. Let f E L° (X, A). 4 (X, A) if and only if there exists a constant C such that, for all n, IITn(f)IIL1 < C. PROOF. 3 with f = 0 yields IITf(f) II tt < 11f 1k' . (f) Tn+1(f). By the Fatou-Beppo Levi theorem, there exists g E L' such that litn Tn(f) = g 36 I. e. Moreover, a direct calculation shows that limTT(f)(x) = f(x) for all x E X. Hence f = g, and therefore f E L'. sup(-f, 0). Then f +, f - E For the general case, set f + = sup(f, 0) and f L°, f +, f - are positive, and f = f + - f -.
Set sp1(x') = limgk(x'), x' E X'. 1, cp1 E M((X', A'); (R, BR)). Furthermore, V(x) = p, (x) if x E X' and o(x) = +oo if x V X'. Let K be a closed subset of R. Then W-1(K) = (pi 1(K) cp1(K)=wi1(K)UG if +oo 0 K if +OOEK. Since cpi 1(K) = X' fl A with A E A and X' E A, it follows that co 1(K) E A. 2 Corollary. M((X,A); (R,%)). Then (limsup M((X, A); (R, Bk)). E 3 Measures and Measure Spaces 13 PROOF. Let (Pn = supp>n fp. Then Vn is measurable. 1 gives the result. Measures and Measure Spaces 3 Definition.
Stochastic integrals (Proc. Durham 1980) by D. Williams