By Howard Raiffa

"In the sector of statistical selection conception, Raiffa and Schlaifer have sought to improve new analytic thoughts through which the fashionable conception of software and subjective likelihood can really be utilized to the industrial research of general sampling problems."

—From the foreword to their vintage paintings *Applied Statistical choice Theory*. First released within the Nineteen Sixties via Harvard college and MIT Press, the booklet is now provided in a brand new paperback variation from Wiley

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Say y(m) →C y(0) if for some k, y(m) j = y(0) j = 0 for all j ≥ k and all m, and y(m)n → y(0)n as m → ∞ for all n. Prove that →C is an L∗ -convergence but that there is no metric e such that y(m) →C y(0) is equivalent to e(y(m), y(0)) → 0. 11. For any two real numbers u and v, max(u, v) := u iff u ≥ v; otherwise, max(u, v) := v. A metric space (S, d) is called an ultrametric space and d an ultrametric if d(x, z) ≤ max(d(x, y), d(y, z)) for all x, y, and z in S. Show that in an ultrametric space, any open ball B(x, r ) is also closed.

B) If (K , d) is perfect, then every non-empty open subset of K is uncountable. 11. Let {xi , i ∈ I } be a net where I is a directed set. For J ⊂ I, {xi , i ∈ J } will be called a strict subnet of {xi , i ∈ I } if J is cofinal in I , that is, for all i ∈ I, i ≤ j for some j ∈ J . (a) Show that this implies J is a directed set with the ordering of I . 1). Hint: Let W be a well-ordering of [0, 1]. Let I be the set of all y ∈ [0, 1] such that {t: t W y} is countable. Show that I is uncountable and well-ordered by W .

By definition of filter base, (F , ≤) is then a directed set. Define a net (using AC) by choosing, for each A ∈ F , an x(A) ∈ A with f (x(A)) ∈ / U . Then the net x(A) → x but f (x(A)) → f (x), contradicting (2). (3) implies (1): take any U ∈ U and x ∈ f −1 (U ). The filter F of all neighborhoods of x converges to x, so f [[F ]] → f (x). For some neighborhood V of x, f [V ] ⊂ U , so V ⊂ f −1 (U ), and f −1 (U ) ∈ T . For another example of a filter base, given a continuous real function f on [0, 1], let t := sup{ f (x): 0 ≤ x ≤ 1}.