An introduction to stochastic processes by D. Kannan PDF

By D. Kannan

ISBN-10: 0444003010

ISBN-13: 9780444003010

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INTEGRALS, BOUNDED, LINEAR FUNCTIONALS, AND MEASURES functions. 2, these are complete and even rings. We call a real-valued function from a Stone vector lattice m a functional, and we call it linear if (i) L(af) = aL(f) (ii) for / e *F, a real number, L ( / 1 + / 2 ) = L(/ 1 ) + L(/ 2 ), nonnegative if (iii) L(f)>0 for |L(/)|0, bounded if (iv) ||/||

1, this functional determines a regular measure X on S. 3. □ If {K(m)} is an increasing sequence of compact sets, we may consider a dense, disjoint, directed sequence {N(m)} of nets such that E{m) = 0 Ef > = K{m\ (3) Let $ be the class of subsets of S that belongs to {iV(m)} as above, and consider a nonnegative, additive set function \i on E. We say that JJ, is tight on the sequence {N{m)} if H(S) - /x(X(m)) -► 0 (m -► oo), (/JL(S) < oo). Since for any positive integer r the sequence of nets with meshes £jm) n K(r\ where m> r, form a dense, disjoint, directed sequence of nets, it follows by the theorem above that \i and the sequence {N(m)} of nets determine a crsmooth measure kr on K(r) for r = 1,2,...

Hence, /*(/) means the integral (6) for a function / , and ju(£) the measure of the set E for a set E. Note that then The integral of a function/ on a set E e £f is defined by jx(f • 1£), also written L f(x)n(dx). This integral has a significance, since / • lE belongs to *F0 if f e ^ 0 . It follows from the definition of the integral that M/-1E1UE2) = M [ ( / - 1 £ 1 ) ] + M / - 1 E 2 ) ? for disjoint sets E1,E2in& , Theorem 2. (7) the functional /x being linear. The set function A, defined on Sf for a given f e*¥0 by X(E) = n(lEf), is a signed measure on £P, and a measure if / > 0.

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An introduction to stochastic processes by D. Kannan

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