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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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 Fall '08
 DavidGarnarnik

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