Algebraic Combinatorics I: Association Schemes by Eiichi Bannai

By Eiichi Bannai

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Lim for every = E Sz = + Fix Define · · · . We first consider the case that In the general case, we use that and apply the construction below to each of + an,d and a positive integer we start by defining the simple function _ . _ = and sets for each integer It is left as j l EJ = { x l ; < f(x) < 1k } ' j, j < k2k . 1 < Then put an exercise to show that the sequence properties. E Let E M . For define the a measurable simple function by Lebesgue integral of s over E {j sdm L Ckm(E E. ). k=l E N = For {sk }f 1 s(x) l fdm = sup D = "Lf 1 Ck XEk(x), we n a measurable, nonnegative function f we define the by over E has all of the desired { l sdm 10 � s � f, ·' Lebesgue integral of f J 11imple .

I n this paper, Hilbert recognized a link being made between integral equations and what we now call linear algebra. Fredholm's 2 This speech has certainly had an impact o n twentieth-century mathematics. An ar­ gument can be made that in addition to its positive impact, it had some negative i nfluence as well. For a discussion of some of the negative effect it has had, see the recent article [52]. 3The friendship and collaboration between Hilbert and Minkowski is quite interesting, but we do not have the space to go into it here.

One way of combining functions is noticeably missing: composition . It is not the case that the composition of two measurable functions is again measurable. See, for example, [1] (page 57 ) or [70] (page 362) to see what can be said about the measurability of the composition of two functions. A real-valued function with only a finite number of elements in its range is called a simple function. One type of simple function is the characteristic function, X E , of a set E s; lR" . This is defined by XE(X ) = { 0I if x e E, if X ;.

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