homework7

Thus corresponds to an impulse train in engineering

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Unformatted text preview: we are led to the measure µ = ∞ δn , that is, µ(A) = ∞ δn (A), for every n=1 n=1 Borel set A. (Thus, µ corresponds to an “impulse train” in engineering parlance. It is also a “counting measure”, in that it just counts the number of integers in a set A.) The statements below are all fairly “obvious” properties of impulses. Your task is to provide a formal proof, being careful to use just the definitions above, the general definition of an integral (as a limit using simple functions), and the property that if two functions are equal except on a set of measure zero, then their integrals are equal. (a) For any nonnegative � (not necessarily simple) measurable function g : R → [0, ∞], we have g dδc = g (c). (b) For any nonnegative (not necessarily simple) measurable function g : � � R → [0, ∞], we have g dµ = ∞ g (n). (This shows that summa­ n=1 tion is a special case of integration.) 1 Exercise 4. (Interchanging summations and limits) Suppose that the numbers aij , ci have the following properties: (i) For every i, the limit limj →∞ aij exists; (ii) For all i, j , we have...
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