Strictly positive measures on Boolean algebras
Strictly positive measures on Boolean algebras
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布尔代数的严格正测度
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
G. Plebanek
中科院分区：
文献类型：
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作者：
M. Džamonja;G. Plebanek
研究了布尔代数上的严格正有限加性测度和紧零维空间上的严格正Radon测度。动机是找到布尔代数的组合特征，布尔代数具有严格正有限加性有限测度，并具有一些附加性质，如可分性或非原子性。Talagrand在1980年提出了一个代数携带可分离严格正测度的可能的一致表征，即该代数的Stone空间K满足其测度的空间M(K)是弱可分离的，等价于C(K)嵌入l∞。我们证明有一个布尔代数的ZFC例子（即紧空间的ZFC例子）满足这个条件，并且不支持严格正测度的可分离性。然而，我们利用这一性质作为一个证明工具，证明了在MA + - CH条件下，每一个大小< c的无原子ccc布尔代数都带有一个非原子严格正测度。算例表明，这一结果在ZFC中不成立。最后，我们得到了带有严格正非原子测度的布尔代数在链式条件下的一个刻画，并得出了在MA + - CH条件下，每个无原子的ccc布尔代数都满足这个强链式条件的结论。
Abstract We investigate strictly positive finitely additive measures on Boolean algebras and strictly positive Radon measures on compact zerodimensional spaces. The motivation is to find a combinatorial characterisation of Boolean algebras which carry a strictly positive finitely additive finite measure with some additional properties, such as separability or nonatomicity. A possible consistent characterisation for an algebra to carry a separable strictly positive measure was suggested by Talagrand in 1980, which is that the Stone space K of the algebra satisfies that its space M(K) of measures is weakly separable, equivalently that C(K) embeds into l∞. We show that there is a ZFC example of a Boolean algebra (so of a compact space) which satisfies this condition and does not support a separable strictly positive measure. However, we use this property as a tool in a proof which shows that under MA + ¬ CH every atomless ccc Boolean algebra of size < c carries a nonatomic strictly positive measure. Examples are given to show that this result does not hold in ZFC. Finally, we obtain a characterisation of Boolean algebras that carry a strictly positive nonatomic measure in terms of a chain condition, and we draw the conclusion that under MA + ¬ CH every atomless ccc Boolean algebra satisfies this stronger chain condition.