By Hurwitz A , Courant R

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O--additive set function A set function ,a:' R* is said to be a--additive (sometimes called completely additive, or countably additive) if (i) Aa(O)=0, (ii) for any disjoint sequence El, E2, ... of sets of such that 00 E _ UEiE'f, i=1 00 p(E) = Z,u(Ei). 3) i=1 As before the condition (i) is redundant if It takes any finite values. Since we may assume that all but a finite number of the sequence {Ei} are void it is clear that any set function which is o--additive is also additive. To see that the converse is not true it is sufficient to consider example (5) on p.

Condition (iii) for p is often called the triangle inequality because it says that the lengths of two sides of a triangle sum to at least that of the third. Condition (ii) ensures that p distinguishes distinct points of X, and (i) says that the distance from y to x is the same as the distance from x to y. When we speak of a metric space X we mean the set X together with a particular p satisfying conditions (i), (ii) and (iii) above. If there is any danger of ambiguity we will speak of the metric space (X, p).

3 2. 4 fail if the set is not closed. To see they also fail if the set is closed but not compact, examine g: R R given by g(x) = exp (x). 3. 4 why could we not have put g = inf {Sx: x A} before first restricting to a finite subset? 4. Suppose A is compact and f f,} is a monotone sequence of continuous functions f,: A -* R converging to a continuous f : A -+ R. Show that the convergence must be uniform, and give an example to show that the condition that A be compact is essential. 5. Prove Lebesgue's covering lemma, which states that if le is an open cover of a compact set A in a metric space (X, p), then there is a 8 > 0, such that the sphere S(x, S) is contained in a set of ' for each x e X.