### A Boolean-valued probability theory

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We define a notion of delta-variance maximization and show it implies epsilon-proximity in expactations.

It is a well-known fact that the Dempster combination rule for combination of uncertainty degrees coming from two or more sources is legitimate only if the combined empirical data, charged with uncertainty and taken as random variables, are statistically (stochastically) independent. We shall prove, however, that for a particular but large enough class of probability measures, an analogy of Dempster combination rule, preserving its extensional character but using some nonstandard and boolean-like...

This contribution introduces the marginal problem, where marginals are not given precisely, but belong to some convex sets given by systems of intervals. Conditions, under which the maximum entropy solution of this problem can be obtained via classical methods using maximum entropy representatives of these convex sets, are presented. Two counterexamples illustrate the fact, that this property is not generally satisfied. Some ideas of an alternative approach are presented at the end of the paper.

Let $(X,\U0001d51b,\mu )$ be a standard probability space. We say that a sub-σ-algebra $\U0001d505$ of $\U0001d51b$decomposes μ in an ergodic way if any regular conditional probability ${}^{\U0001d505}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}P$ with respect to $\U0001d505$ andμ satisfies, for μ-almost every x∈X, $\forall B\in \U0001d505,{}^{\U0001d505}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}P(x,B)\in \{0,1\}$. In this case the equality $\mu (\xb7)={\int}_{X}{}^{\U0001d505}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}P(x,\xb7)\mu \left(\mathrm{d}x\right)$, gives us an integral decomposition in “$\U0001d505$-ergodic” components. For any sub-σ-algebra $\U0001d505$ of $\U0001d51b$, we denote by $\overline{\U0001d505}$ the smallest sub-σ-algebra of $\U0001d51b$ containing $\U0001d505$ and the collection of all setsAin $\U0001d51b$ satisfyingμ(A)=0. We say that $\U0001d505$ isμ-complete if $\U0001d505=\overline{\U0001d505}$. Let $\{{\U0001d505}_{i}i\in I\}$ be a non-empty family...