We prove that if \(\mathcal {A}\) is an infinite Boolean algebra in the ground model V and \(\mathbb {P}\) is a notion of forcing adding any of the following reals: a Cohen real, an unsplit real, or a random real, then, in any \(\mathbb {P}\)-generic extension V[G], \(\mathcal {A}\) has neither the Nikodym property nor the Grothendieck property. A similar result is also proved for a dominating real and the Nikodym property.

中文翻译：

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添加真实值后措施的收敛

我们证明，如果\(\mathcal {A}\)是基础模型V中的无限布尔代数，并且\(\mathbb {P}\)是强制添加以下任意实数的概念：科恩实数、那么，在任何\(\mathbb {P}\)泛型扩展V [ G ] 中，\(\mathcal {A}\)既不具有 Nikodym 性质，也不具有 Grothendieck 性质。对于支配实数和 Nikodym 属性也证明了类似的结果。