Borels Theorem

Consider the two functions F,(t) and F2(t) whose Laplace transforms are given as flip) and/ (p) respectively. Then the product 

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Theorem 5.12. A nonnegative measure can he defined on the a-algebra of Borel subsets of I such that the representation... 

One additional result is needed, and it is derived in Appendix 2 of this chapter. This is Borel s theorem, which states ... 

This is the solution of the problem in transform space. We have a direct relationship between the transforms of the compliance and the modulus. This solution must now be returned to real space. Making use of Borel s theorem, equation (k), and the result derived in equation (e) gives the final result ... 

In these more complicated examples, we are able to demonstrate ergodicity without finding an explicit solution (as in the Ornstein-Uhlenbeck example) or study of the dynamics generator (as in Brownian dynamics), given some assumptions on the behavior of solutions. We state (without proof) a powerful theorem on the ergodicity of degenerate stochastic diffusions, whose proof is essentially contained in [257, Theorem 2.5] (see also [44, 160, 161, 253, 266]). We denote by Hfix) the open ball in D centered on the points of radius while B(D) is the Borel a-algebra on T) (see Sect. 5.2.1). 

Theorem 4.1.7 (Trofimov [130]-[133]). Let G be a simple Lie algebra of one of the following types so(n),su(n),sp(n),G2. Then on each orbit of general position in the real form of the Borel (solvable) subalgebra BG (of the algebra G) there always exists a maximal linear commutative algebra of polynomials. These polynomials are written by explicit formulae. 

Theorem 4.1.9. Let BG be a Borel subalgebra in a semisimple complex Lie algebra G. Then on BG there always exists a maximal commutative linear algebra of polynomials. 

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