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let x s micro be a measure space and s 1middot1 a separable banach space a function f from x into s will be called
1 let s d be a metric space e sub s and f a complex-valued lipschitz function defined on e with f x - f y le kdx y
1 prove that a banach space x is reflexive if and only if its dual x 1 is reflexive note only if is easier2 let x
let f be a lipschitzian function on x and e a subset of x where f l on x is the same as for its restriction to e
for a finite measure space x s micro a set f of integrable functions is said to be uniformly integrable iff sup f
1 let f be a measurable function from x onto s where x a is a mea- surable space and s e is a metric space with borel
1 let x1 x2 be iid real random variables with e x1 lt gtinfin and sn x1 middot middot middot xn verify that sn
as in problem 4 in sect101 let x1 x2 be iid real random variables with e x1 infin let sn x1 middot middot middot xn
let an be an increasing sequence of sigma-algebras whose union generates a sigma-algebra s let micro and nu be
1 a player throws a fair coin and wins 1 each time its heads but loses 1 each time its tails the player will stop
show that for any metric space s if a sub s and x isin a a then there is a bounded continuous real-valued function on a
1 let x d be a locally compact separable metric space show that its one-point compactification is metrizable2 let x be
let k be the tychonoff-c ech compactification of r show that addition from r times r onto r cannot be extended to a
if x t is a locally compact hausdorff space show that x as a subset of its tychonoff-c ech compactification k is
given a uniform space s u a net xalpha alphaisini in s is called a cauchy net iff for any v isin u there is a gamma
a uniform space s u is called separated iff for every x y in s there is a u isin u with x y isin u a show that the
1 if x y and z are uniform spaces f is uniformly continuous from x into y and g is uniformly continuous from y into
1 a topological space s t is called perfectly normal iff for every closed set f there is a continuous real function f
1 a show that any open set u in r is a union of countably many disjoint open intervals one or two of which may be
1 show that the set rq of irrational numbers with usual topology relative topology from r is topologically complete2
in the proof of proposition 1211 a 1-1 measurable function f from r onto 0 1 was constructed by representing each
let xt be a brownian motion show that for any epsilongt 0 and almost all omega there is an m omega infin such that for
for iid variables x1 x2 let an be the smallest sigma -algebra for which x1 xn are measurable and bn the smallest sigma
1 reconsider the continuous-time markov chain example discussed in class assume now that the machines are not serially
let q run through the collection of all laws for which eq f 2 infin for the u -statistic un show that a un is the