The representation of σ-algebras
The representation of σ-algebras
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σ-代数的表示
DOI：
10.1007/978-1-4612-9855-7_23
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发表时间：
1974
影响因子：
1.5
通讯作者：
P. Halmos
中科院分区：
文献类型：
--
作者：
P. Halmos
我们知道每一个布尔代数都同构于一个域，而一个完备的布尔代数不需要同构于一个完备的域（因为它不需要是原子的）。很自然地会问一个中间问题：每个σ-代数是否同构于一个σ-域？事实上，我们将看到，如果A是满足可数链条件的非原子σ-代数，则A不能同构于σ-域。对于这样的代数的一个例子，考虑没有孤立点的Hausdorff空间的正则开代数和可数基。或者，考虑单位区间的约化Borel代数或约化测度代数。
We know that every Boolean algebra is isomorphic to a field, whereas a complete Boolean algebra need not be isomorphic to a complete field (since, for instance, it need not be atomic). It is natural to ask the intermediate question: is every σ-algebra isomorphic to a σ-field? The answer is no. We shall see, in fact, that if A is a non-atomic σ-algebra satisfying the countable chain condition, then A cannot be isomorphic to a σ-field. For an example of such an algebra consider the regular open algebra of a Hausdorff space with no isolated points and with a countable base. Alternatively, consider either the reduced Borel algebra or the reduced measure algebra of the unit interval.