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Page 45
H. L. Royden. 17. Proposition : Let f be a real - valued function defined and con- tinuous on a closed and bounded set F. Then f is bounded on F and assumes its maximum and minimum on F ; that is , there are points x1 and x2 in F such that ...
H. L. Royden. 17. Proposition : Let f be a real - valued function defined and con- tinuous on a closed and bounded set F. Then f is bounded on F and assumes its maximum and minimum on F ; that is , there are points x1 and x2 in F such that ...
Page 47
H. L. Royden. 42 73 Problems 39. Let F be a closed set of real numbers and ƒ a real - valued function which is defined and continuous on F. Show that there is a function g defined and continuous on ( -∞ , ∞ ) such that f ( x ) = g ( x ) ...
H. L. Royden. 42 73 Problems 39. Let F be a closed set of real numbers and ƒ a real - valued function which is defined and continuous on F. Show that there is a function g defined and continuous on ( -∞ , ∞ ) such that f ( x ) = g ( x ) ...
Page 150
... f on X such that 0 ≤f≤ 1 , f = 0 on A and ƒ = 1 on B. 19. Prove Tietze's Extension Theorem by using the following steps : a . Let h = f / ( 1 + | ƒ | ) . Then | h | < 1 . > b . Let B = { x : h ( x ) ≤ - } } , C = { x : h ( x ) ≥ 3 ) ...
... f on X such that 0 ≤f≤ 1 , f = 0 on A and ƒ = 1 on B. 19. Prove Tietze's Extension Theorem by using the following steps : a . Let h = f / ( 1 + | ƒ | ) . Then | h | < 1 . > b . Let B = { x : h ( x ) ≤ - } } , C = { x : h ( x ) ≥ 3 ) ...
Contents
Prologue to the Student | 1 |
Topological Spaces | 4 |
Measure and Outer Measure | 12 |
Copyright | |
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A₁ absolutely continuous axiom Baire measure Baire set Banach space Borel measure Borel sets bounded linear functional called Cauchy sequence closed sets compact Hausdorff space compact space continuous function continuous real-valued function Convergence Theorem countable collection definition denote E₁ element finite measure finite number fn(x following proposition function defined function f ƒ and g ƒ is continuous given Hausdorff space Hence homeomorphism integrable function L₁ Lebesgue measure Lemma Let f Let ƒ locally compact measurable function measurable sets measure space measure zero monotone natural numbers nonempty nonnegative measurable function o-algebra o-finite one-to-one open intervals open set outer measure point of closure Problem Proof rational numbers semicontinuous sequence of measurable set containing set function set of finite set of measure simple functions subspace topological space topological vector space unique