By Benz W.

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If f : D → is a non-negative function, we define its integral as f dµ = sup g dµ : g is simple, 0 g f . Notes on probability theory 17 To complete the definition, if f takes both positive and negative values, we let f+ (x) = max{f (x), 0} and f− (x) = max{−f (x), 0}, so that f = f+ − f− , and define f dµ = f+ dµ − f− dµ provided that f+ dµ and f− dµ are both finite. All the usual properties hold for integrals, for example, (f + g)dµ = f dµ + g dµ and λf dµ = λ f dµ if λ is a scalar. We also have the monotone convergence theorem, that if fk : D → is an increasing sequence of non-negative functions converging (pointwise) to f , then lim k→∞ fk dµ = f dµ.

A transformation T : n → n is linear if T (x + y) = T (x) + T (y) and T (λx) = λT (x) for all x, y ∈ n and λ ∈ ; linear transformations may be represented by matrices in the usual way. Such a linear transformation is nonsingular if T (x) = 0 if and only if x = 0. If S : n → n is of the form S(x) = T (x) + a, where T is a non-singular linear transformation and a is a point in n , then S is called an affine transformation or an affinity. An affinity may be thought of as a shearing transformation; its contracting or expanding effect need not be the same in every direction.

Moreover, if F is compact, then, by expanding the covering sets slightly to open sets, and taking a finite subcover, we get the same value of Hs (F ) if we merely consider δ-covers by finite collections of sets. Net measures are another useful variant. For the sake of simplicity let F be a subset of the interval [0, 1). A binary interval is an interval of the form [r2−k , (r + 1)2−k ) where k = 0, 1, 2, . . and r = 0, 1, . . , 2k − 1.

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