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let x s be a measurable space and en measurable sets not necessarily disjoint whose union is x suppose that for each n
show that for some topological spaces x s and y t there is a closed set d in x times y with product topology which is
let s t be a second-countable topological space and y d any metric space show that the borel sigma-algebra in the
let r be a sigma-ring of subsets of a set x let s be the sigma-algebra generated by r recall sect33 problem 8 or prove
let y be a random variable independent of all the x j with e y ltgtinfin let uj y tj for j -1 -2 let ck be the
let x1 x2 be independent real random variables with exn 0 for all n and sn x1 middotmiddotmiddot xn let bn be the
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
1 let x y and z be independent real random variables with eex infin e x infin ey 0 and ez 2 infin show that x ex y
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
9 let g be the affine group of the line namely the set of all transformations x frarr ax b of r onto itself where a
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 let s d be any noncompact metric space show that there exist bounded continuous functions fn on s such that fn x
a c 1 curve is a function t frarr f t gt from r into r2 where the derivatives f 1t and g1t exist and are
1 show that the set rq of irrational numbers with usual topology relative topology from r is topologically complete2