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Can sigma field be countably infinite? - Answers

No. The basic idea of proving why not is this: 1) The underlying space X is at least countably infinite (of course). 2) Use the properties of a sigma field (aka sigma algebra) to find a countable partition of the space, X = disjiont-union( X_i ). 3) Notice that the union(X_i, s in S) is in the sigma algebra for any subset S of natural numbers. 4) Notice that any union(X_i, s in S) is distinct. 5) Conclude, since the set of subsets of natural numbers is uncountable, so too is your sigma algebra.



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Can sigma field be countably infinite? - Answers

https://math.answers.com/math-and-arithmetic/Can_sigma_field_be_countably_infinite

No. The basic idea of proving why not is this: 1) The underlying space X is at least countably infinite (of course). 2) Use the properties of a sigma field (aka sigma algebra) to find a countable partition of the space, X = disjiont-union( X_i ). 3) Notice that the union(X_i, s in S) is in the sigma algebra for any subset S of natural numbers. 4) Notice that any union(X_i, s in S) is distinct. 5) Conclude, since the set of subsets of natural numbers is uncountable, so too is your sigma algebra.



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https://math.answers.com/math-and-arithmetic/Can_sigma_field_be_countably_infinite

Can sigma field be countably infinite? - Answers

No. The basic idea of proving why not is this: 1) The underlying space X is at least countably infinite (of course). 2) Use the properties of a sigma field (aka sigma algebra) to find a countable partition of the space, X = disjiont-union( X_i ). 3) Notice that the union(X_i, s in S) is in the sigma algebra for any subset S of natural numbers. 4) Notice that any union(X_i, s in S) is distinct. 5) Conclude, since the set of subsets of natural numbers is uncountable, so too is your sigma algebra.

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      No. The basic idea of proving why not is this: 1) The underlying space X is at least countably infinite (of course). 2) Use the properties of a sigma field (aka sigma algebra) to find a countable partition of the space, X = disjiont-union( X_i ). 3) Notice that the union(X_i, s in S) is in the sigma algebra for any subset S of natural numbers. 4) Notice that any union(X_i, s in S) is distinct. 5) Conclude, since the set of subsets of natural numbers is uncountable, so too is your sigma algebra.

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