By Luis Garrido

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**Extra info for Fluctuations and Stochastic Phenomena in Condensed Matter**

**Example text**

B) Let g be a positive simple function such that f ≥ g. 10 yields i = 1, . . , m. 11 lim μfn = lim n n i=1 m lim μ(fn 1Bi ) ≥ μ(fn 1Bi ) = i=1 n bi μ(Bi ) = μg. 3b. For each k, the function dk ◦ f is simple and f ≥ dk ◦ f . 11, we have lim μfn ≥ μ(dk ◦ f ) n for all k. Letting k lim μfn ≥ μf . → ∞ we obtain the desired inequality that 24 Measure and Integration Chap. 12 Proposition. For f and g in E+ and a and b in R+ , μ(af + bg) = a μf + b μg. The same is true for integrable f and g in E and arbitrary a and b in R.

F ⇒ L(fn ) L(f ). (fn ) ⊂ E+ and fn Proof. Necessity of the conditions is immediate from the properties of the integral: (a) follows from the deﬁnition of μf , (b) from linearity, and (c) from the monotone convergence theorem. To show the suﬃciency, suppose that L has the properties (a)-(c). 23 A ∈ E. μ(A) = L(1A ), We show that μ is a measure. First, μ(∅) = L(1∅ ) = L(0) = 0. Second, if n A1 , A2 , . . are disjoint sets in E with union A, then the indicator of 1 Ai is n 1 1Ai , the latter is increasing to 1A , and hence, n μ(A) = L(1A ) = lim L( n n 1Ai ) = lim n 1 ∞ n L(1Ai ) = lim n 1 μ(Ai ) = 1 μ(Ai ), 1 where we used the conditions (c) and (b) to justify the second and third equality signs.

30 Transition densities. Let ν be a σ-ﬁnite measure on (F, F), and let k be a positive function in E ⊗ F. 2, that is, in diﬀerential notation, K(x, dy) = ν(dy) k(x, y). Show that K is a transition kernel. Then, k is called the transition density function of K with respect to ν. 31 Finite spaces. Let E = {1, . . , m}, F = {1, . . , n}, G = {1, . . , p} with their discrete σ-algebras.