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Let F be a collection of subsets of a set X. Then the intersection A of all o-algebras of subsets of X that contain F is a o-algebra that contains F. Moreover, it is the smallest o-algebra of subsets of X that contains F in the sense that any o-algebra that contains F also contains A.

Let F be a collection of subsets of a set X. Then the intersection A of all o-algebras of subsets of X that contain F is a o-algebra that contains F. Moreover, it is the smallest o-algebra of subsets of X that contains F in the sense that any o-algebra that contains F also contains A.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let F be a collection of subsets of a set X. Then the intersection A of all o-algebras of subsets of X that contain F is a o-algebra that contains F. Moreover, it is the smallest o-algebra of subsets of X that contains F in the sense that any o-algebra that contains F also contains A. Solution: Let E represent the collection of all o-algebras of X that contain F as a subset. Meaning, FCB for all Be E. By hypothesis, A = N B. BeE Show that FCA.
Transcribed Image Text:Let F be a collection of subsets of a set X. Then the intersection A of all o-algebras of subsets of X that contain F is a o-algebra that contains F. Moreover, it is the smallest o-algebra of subsets of X that contains F in the sense that any o-algebra that contains F also contains A. Solution: Let E represent the collection of all o-algebras of X that contain F as a subset. Meaning, FCB for all Be E. By hypothesis, A = N B. BeE Show that FCA.
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