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1 take the cantor function g defined in the proof of proposition 421 as a nondecreasing function it defines a measure
1 if micro and nu are finite measures on a sigma-algebra s show that nu is absolutely continuous with respect to micro
1 let fnk x ank for 2k le 2n x lt gt2k 1 and fnk x bnk for 2k 1 le x 2k 2 for each n 0 1 and k 0 1 k
recall that a radon measure on r is a function micro into r defined on all bounded borel sets and countably additive on
1 let f x x p for all real x for what values of p is it true that a f is convex b the derivative f ix exists for
1nbsp for any set f and point x in a metric space recall that dx f infdx y y isin f let f be a
1 suppose that a real-valued function f on an open interval j in r has a second derivative f ii on j show that f is
1 show that in any finite dimensional banach space rk with any norm for any closed convex set c and any point x not in
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