- #1

Bacle

- 662

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Hi, All:

A simple question: If A is the sigma- algebra generated by a collection C of subsets

of an ambient set X. Isn't it trivial that the sigma-algebra generated by A is A itself?

One definition is that the sigma algebra generated by a collection S of subsets is the

intersection of all s-algebras containing the collection S . Doesn't this automatically

mean that A is thesigma algebra generated by A? I am trying to avoid using the fact

that a countable union of countable sets is countable (otherwise, each set in A is the

countable union of sets in C, so a countable union of subsets of A is a countable union

of a countable union of sets in C , which is itself a countable union, so it is

contained in A, by def. of A as the s-algebra generated by C --a tongue-twister and proof! )

Thanks.

P.S: I am trying to answer the question for a student who has not yet seen cardinal arithmeti;

that is why I am trying to avoid using that union of countable is countable,

A simple question: If A is the sigma- algebra generated by a collection C of subsets

of an ambient set X. Isn't it trivial that the sigma-algebra generated by A is A itself?

One definition is that the sigma algebra generated by a collection S of subsets is the

intersection of all s-algebras containing the collection S . Doesn't this automatically

mean that A is thesigma algebra generated by A? I am trying to avoid using the fact

that a countable union of countable sets is countable (otherwise, each set in A is the

countable union of sets in C, so a countable union of subsets of A is a countable union

of a countable union of sets in C , which is itself a countable union, so it is

contained in A, by def. of A as the s-algebra generated by C --a tongue-twister and proof! )

Thanks.

P.S: I am trying to answer the question for a student who has not yet seen cardinal arithmeti;

that is why I am trying to avoid using that union of countable is countable,

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