Iterations of Boolean algebras with measure. (English) Zbl 0687.03032

Kamburelis, Anastasis

Arch. Math. Logic 29, No. 1, 21-28 (1989).

The author considers a class M of Boolean algebras with strictly positive, finitely additive measures. He shows that M is closed under iteration with finite support and that forcing via such algebras allows an extension of the Lebesgue measure structure of the ground model. For a class \(\Gamma\) of Boolean algebras let MA(\(\Gamma)\) denote Martin’s axiom restricted to \(\Gamma\). B is said to be \(\sigma\)-centered if the Stone space of B is separable. The main result of this paper is: MA(M)\(\equiv MA(\sigma\)-centered) and LC, where LC is the statement that for an infinite cardinal \(\kappa\), \(Cov(L_{\kappa})=2^{\omega}\), with a suitable definition of \(Cov(L_{\kappa})\).

Reviewer: Mo Shaokui

MSC:

03E40	Other aspects of forcing and Boolean-valued models
03E50	Continuum hypothesis and Martin’s axiom
28A60	Measures on Boolean rings, measure algebras

Keywords:

measures on Boolea algebras; forcing; Martin’s axiom

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