In his 1998 Fubini Lectures, Rota discusses twelve problems in probability that “no one likes to bring up”. The first problem calls for a revision of the notion of a sample space, guided by the belief that mention of sample points in a probabilistic argument is bad form and that a “pointless” foundation of probability should be provided by algebras of random variables.

In 1958 Chang introduced MV-algebras to prove the completeness theorem of Łukasiewicz logic Ł_∞. The aim of this paper is to show that MV-algebras provide a solution of Rota's first problem.

The adjunction between MV-algebras and unital commutative C*-algebras equips every MV-algebra A with a natural ring structure, as advocated by Nelson for algebras of random variables. The closed compact set $\mathsf{S}(A)\subseteq {\mathrm{[}\mathrm{0},\mathrm{1}\mathrm{]}}^A$ of finitely additive probability measures on A (the states of A) coincides with the set of $\mathrm{[}\mathrm{0},\mathrm{1}\mathrm{]}$-valued functions on A whose finite restrictions are consistent in de Finetti's sense. MV-algebras and Ł_∞ thus provide the framework for a generalization (known as ŁIPSAT) of Boole's probabilistic inference problem, and its modern reformulation known as probabilistic satisfiability, PSAT. We construct an affine homeomorphism ${\gamma }_A$ of $\mathsf{S}(A)$ onto the weakly compact space of regular Borel probability measures on the maximal spectral space $\mathit{\mu }(A)$. The latter is the most general compact Hausdorff space. As a consequence, for every Kolmogorov probability space $(\mathrm{\Omega },{\mathcal{F}}_{\mathrm{\Omega }},P)$, with ${\mathcal{F}}_{\mathrm{\Omega }}$ the sigma-algebra of Borel sets of a compact Hausdorff space Ω, and P a regular probability measure on ${\mathcal{F}}_{\mathrm{\Omega }}$, there is an MV-algebra A and a state σ of A such that $(\mathrm{\Omega },{\mathcal{F}}_{\mathrm{\Omega }},P)$ $\cong (\mathit{\mu }(A),{\mathcal{F}}_{\mathit{\mu }(A)},{\gamma }_A(\sigma ))$.

在他的1998年Fubini讲座中，Rota讨论了“没人喜欢提出”的十二个问题。第一个问题要求对样本空间的概念进行修订，并应相信以下观点：概率论证中提到样本点是不好的形式，而概率的“无意义”基础应该由随机变量的代数提供。

1958年，Chang引入MV-代数以证明Łukasiewicz逻辑_∞的完备性定理。本文的目的是证明MV代数提供了Rota第一个问题的解决方案。

MV代数和单位可交换C *代数之间的结合使每个MV代数A都具有自然的环结构，这是纳尔逊（Nelson）提倡的随机变量代数。密闭套装$\mathsf{小号}（\mathrm{一种}）\subseteq {\mathrm{[}\mathrm{0}，\mathrm{1个}\mathrm{]}}^{\mathrm{一种}}$对有限可加概率测度甲（的状态甲）与该组的一致$\mathrm{[}\mathrm{0}，\mathrm{1个}\mathrm{]}$A的有限值函数在de Finetti的意义上是一致的。MV代数和L _∞从而为布尔的概率推理问题的推广（被称为ŁIPSAT）的框架，它的现代再形成被称为概率满足性，PSAT。我们构造一个仿射同胚${\gamma }_{\mathrm{一种}}$ 的 $\mathsf{小号}（\mathrm{一种}）$ 最大频谱空间上常规Borel概率测度的弱紧空间 $\mathit{\mu }（\mathrm{一种}）$。后者是最普通的紧凑型Hausdorff空间。结果，对于每个Kolmogorov概率空间$（\mathrm{\Omega }，{\mathcal{F}}_{\mathrm{\Omega }}，P）$，带有 ${\mathcal{F}}_{\mathrm{\Omega }}$紧Hausdorff空间Ω的Borel集的σ代数，P是${\mathcal{F}}_{\mathrm{\Omega }}$，还有一个MV-代数甲和状态σ的甲，使得$（\mathrm{\Omega }，{\mathcal{F}}_{\mathrm{\Omega }}，P）$ $\cong （\mathit{\mu }（\mathrm{一种}），{\mathcal{F}}_{\mathit{\mu }（\mathrm{一种}）}，{\gamma }_{\mathrm{一种}}（\sigma ））$。