By Thomas P. Wangler

ISBN-10: 0471168149

ISBN-13: 9780471168140

The 1st e-book that offers a unmarried resource of introductory info an all linear accelerators, together with electron and ion accelerators.

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**Additional info for Principles of RF linear accelerators**

**Sample text**

14. Take a topological space X with a topology T. ) If X is a metric space equipped with the metric topology T, then it has a base of open balls (every open set is the union of open balls). A topological space is separable if it has a countable dense subset. A metric space is separable if and only if it has a countable base of open balls. Show that the σ-algebra XT generated by the metric topology T on a separable metric space coincides with the σ-algebra XB generated by any countable base B of open balls.

The sets in T are called the open sets of X with respect to T. Let X be equipped with a topology, and equip the real line R with its usual topology (the one induced by the usual metric on R ). Show that if a real-valued function f : X → R is continuous (with respect to the above topologies, which means that the inverse image under f of open sets of R are open sets of X), then f is measurable with respect to the σ-algebra XT generated by a topology T on X. 1 this means that continuous functions are Borel measurable).

Suppose {fn } converges μ-almost everywhere to f , so that {fn } converges pointwise to f on E = X\N for some N ∈ X with μ(N ) = 0. Thus {fn χE } is an increasing sequence of functions in M(X, X )+ converging pointwise to f χE ∈ M(X, X )+. By the Monotone Convergence Theorem, f χE dμ = lim n fn χE dμ. 50 3. 8) that f χE dμ = f dμ = f dμ and fn χE dμ = E fn dμ = fn dμ.

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