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added link https://en.wikipedia.org/wiki/Maharam%27s_theorem

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edit approved Apr 16, 2021 at 18:42

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You write "An example of a σ-algebra that has no atoms but supports a probability measure is $\{0,1\}^\kappa$ for $\kappa$ uncountable, which we can endow with the coin-flipping probability measure."

Maharam's theoremMaharam's theorem says that these are essentially the only ones. That is: Every Boolean algebra which is equipped with a probability measure (and is Dedekind complete, see below) is isomorphic to a product of the measure algebras on various $2^\kappa$ that you mentioned. (Including finite $\kappa$, to take care of measures with atoms.)

Dedekind complete means that every subset has a least upper bound. If you take a $\sigma$-algebra which carries a $\sigma$-additive probability measure, and divide by the ideal of null sets, then the resulting algebra is still a measure algebra and it will be Dedekind complete.

An exposition of Maharam's theorem can be found in Fremlin's book, volume 3. (The theorem I quoted can be generalized to algebras with a "semifinite" measure, which is more general than probability measure.)

You write "An example of a σ-algebra that has no atoms but supports a probability measure is $\{0,1\}^\kappa$ for $\kappa$ uncountable, which we can endow with the coin-flipping probability measure."

Maharam's theorem says that these are essentially the only ones. That is: Every Boolean algebra which is equipped with a probability measure (and is Dedekind complete, see below) is isomorphic to a product of the measure algebras on various $2^\kappa$ that you mentioned. (Including finite $\kappa$, to take care of measures with atoms.)

Dedekind complete means that every subset has a least upper bound. If you take a $\sigma$-algebra which carries a $\sigma$-additive probability measure, and divide by the ideal of null sets, then the resulting algebra is still a measure algebra and it will be Dedekind complete.

An exposition of Maharam's theorem can be found in Fremlin's book, volume 3. (The theorem I quoted can be generalized to algebras with a "semifinite" measure, which is more general than probability measure.)

You write "An example of a σ-algebra that has no atoms but supports a probability measure is $\{0,1\}^\kappa$ for $\kappa$ uncountable, which we can endow with the coin-flipping probability measure."

Maharam's theorem says that these are essentially the only ones. That is: Every Boolean algebra which is equipped with a probability measure (and is Dedekind complete, see below) is isomorphic to a product of the measure algebras on various $2^\kappa$ that you mentioned. (Including finite $\kappa$, to take care of measures with atoms.)

Dedekind complete means that every subset has a least upper bound. If you take a $\sigma$-algebra which carries a $\sigma$-additive probability measure, and divide by the ideal of null sets, then the resulting algebra is still a measure algebra and it will be Dedekind complete.

An exposition of Maharam's theorem can be found in Fremlin's book, volume 3. (The theorem I quoted can be generalized to algebras with a "semifinite" measure, which is more general than probability measure.)

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created Jan 17, 2012 at 11:56

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You write "An example of a σ-algebra that has no atoms but supports a probability measure is $\{0,1\}^\kappa$ for $\kappa$ uncountable, which we can endow with the coin-flipping probability measure."

Maharam's theorem says that these are essentially the only ones. That is: Every Boolean algebra which is equipped with a probability measure (and is Dedekind complete, see below) is isomorphic to a product of the measure algebras on various $2^\kappa$ that you mentioned. (Including finite $\kappa$, to take care of measures with atoms.)

Dedekind complete means that every subset has a least upper bound. If you take a $\sigma$-algebra which carries a $\sigma$-additive probability measure, and divide by the ideal of null sets, then the resulting algebra is still a measure algebra and it will be Dedekind complete.

An exposition of Maharam's theorem can be found in Fremlin's book, volume 3. (The theorem I quoted can be generalized to algebras with a "semifinite" measure, which is more general than probability measure.)