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Let 2 be a non-empty set. Let Fo be the collection of all subsets such that either A or AC is finite. (a) Show that Fo is a field. Define for E E Fo the set function P by 0, P(E) = { if E is finite, if EC is finite. 1, (b) If S2 is countably infinite, show P is finitely additive but not o-additive. (c) If S2 is uncountable, show P is o-additive on Fo.

Let 2 be a non-empty set. Let Fo be the collection of all subsets such that either A or AC is finite. (a) Show that Fo is a field. Define for E E Fo the set function P by 0, P(E) = { if E is finite, if EC is finite. 1, (b) If S2 is countably infinite, show P is finitely additive but not o-additive. (c) If S2 is uncountable, show P is o-additive on Fo.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter7: Real And Complex Numbers
Section7.1: The Field Of Real Numbers
Problem 21E: Let S be a nonempty subset of an order field F. Write definitions for lower bound of S and greatest...
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Let 2 be a non-empty set. Let Fo be the collection of all subsets such that either A or AC is finite. (a) Show that Fo is a field. Define for E E Fo the set function P by 0, if E is finite, P(E) = { 1, if EC is finite. (b) If S2 is countably infinite, show P is finitely additive but not o-additive. (c) If S2 is uncountable, show P is o-additive on Fo.
Transcribed Image Text:Let 2 be a non-empty set. Let Fo be the collection of all subsets such that either A or AC is finite. (a) Show that Fo is a field. Define for E E Fo the set function P by 0, if E is finite, P(E) = { 1, if EC is finite. (b) If S2 is countably infinite, show P is finitely additive but not o-additive. (c) If S2 is uncountable, show P is o-additive on Fo.
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