Real Analysis |
From inside the book
Results 1-3 of 86
Page 20
... one - to - one correspondence with a countable set must be countable . Since the set N of natural numbers is ... each n in N has a unique factorization of the form3 n = 21 32 ... p , where x ; e No = NU { 0 } and xx > 0 . Let f be the ...
... one - to - one correspondence with a countable set must be countable . Since the set N of natural numbers is ... each n in N has a unique factorization of the form3 n = 21 32 ... p , where x ; e No = NU { 0 } and xx > 0 . Let f be the ...
Page 33
... one - to - one . We can also prove by induction that ( p + q ) = ❤ ( p ) + ( q ) and ❤ ( pq ) = ❤ ( p ) ¥ ( q ) . Thus gives a one - to - one correspondence between the natural numbers and a subset of R , and preserves sums ...
... one - to - one . We can also prove by induction that ( p + q ) = ❤ ( p ) + ( q ) and ❤ ( pq ) = ❤ ( p ) ¥ ( q ) . Thus gives a one - to - one correspondence between the natural numbers and a subset of R , and preserves sums ...
Page 330
... one - to - one on ❤ Y ~ Yo , and 1 [ A ] = Þ ( A ) modulo Л , [ B ] Thus [ Y ~ Yo ] differs from X by a set Xo e M. □ = ( B ) modulo M. Unfortunately , the exceptional sets Xo and Yo in the preceding theorem may be unavoidable ( see ...
... one - to - one on ❤ Y ~ Yo , and 1 [ A ] = Þ ( A ) modulo Л , [ B ] Thus [ Y ~ Yo ] differs from X by a set Xo e M. □ = ( B ) modulo M. Unfortunately , the exceptional sets Xo and Yo in the preceding theorem may be unavoidable ( see ...
Contents
Prologue to the Student | 1 |
Topological Spaces | 4 |
Measure and Outer Measure | 12 |
Copyright | |
64 other sections not shown
Other editions - View all
Common terms and phrases
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