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Sigma algebras, measure spaces, rectangles, and cylinders

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Let (Omega_1, F_1, P_1) and (Omega_2, F_2, P_2) be the following measure spaces:

Omega_1 = {a, b}, F_1 is the sigma algebra of all subsets of Omega_1, and P_1 is a measure on Omega_1.

Omega_2 = {c, d}, F_2 is the sigma algebra of all subsets of Omega_2, and P_2 is a measure on Omega_2.

Determine the makeup of the set C of cylinders of Omega_1 and the set R of rectangles of Omega_2, and show that C is a subset of R.

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Solution Summary

The set of rectangles and the set of cylinders for the given pair of measure spaces is determined (in detail), and it is shown that the set of cylinders is contained in the set of rectangles.

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Let Omega_1 = {a, b} and Omega_2 = {c, d}.

By definition of the sigma algebra of all subsets of Omega_1,

F_1 = {empty set, {a}, {b}, {a,b}}

Similarly, by definition of the sigma algebra of all ...

Solution provided by:
  • AB, Hood College
  • PhD, The Catholic University of America
  • PhD, The University of Maryland at College Park
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