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Additional resources for Algebra and number theory: proceedings of a conference held in Fez, Morocco
8). B'-measurable mapping of 'D(/). 1 r Proposition 1A2. By is the Borel a-algebra of subsets of Y. Let f be a mapping with t;D(f) E 2l and ~(f) c Y. Let Dy and (;y be respectively the collection of all open sets and the collection of all closed sets in Y. B'y-measurable mapping oft;D(f) into Y if and only if f- 1(Dy) C 2l. B'y-measurable mapping uft;D(f) into Y if and only if f- 1(((;y) C 2l. Proof. 41. 43. By are the Borel CT-algebras of subsets of X and Y respectively. B'y-measurable mapping of D into Y.
On 2l. ) If fl. *( lim En)· Weshow n-+oo n--+oo next that if fl. • is a regular outer measure then the equality holds. 11. * be an outer measure on a set X and let (En : n E N) be an increasing sequence of subsets of X. L* is a regular outer measure then (2) Proof. 1. Let (E11 : n E N) be an increasing sequence of subsets of X. L"'(E11 ) exists. ). This proves (1). oo 2. L* is a regular outer measure. L*). Now En C Fn for n EN implies that IiminfEn C liminfF,.. L*(En). This and 11--+00 (1) imply (2).
By are the Borel CT-algebras of subsets of X and Y respectively. B'y-measurable mapping of D into Y. 22 CHAPTER 1 Measure Spaces Proof. Let V be an open set in Y. 42. 43 is when we have a real-valued continuous function defined on a set D E ~ x where ~ x is the Borel a-algebra of subsets of a topological space X. In this case we have (Y, ~y) = (JR, ~JR). 43, f is a ~x/~JR measurable mapping of D into JR. f [IX] Induction of Measure by Measurable Mapping Let JL be a measure on a a-algebra~ of subsets of a set X.