On p. 48 in Troffaes and de Cooman's Lower Previsions there's a claim that "It is clear that" the natural extension $\underline{E}_\underline{P}$ of lower prevision $\underline{P}$ dominates (42) it, i.e. that $\underline{E}_\underline{P}(f) \geq \underline{P}(f)$ for all bounded functions ("gamb...

In Example 11.6 on p. 270 of Hermans and Škulj's "Stochastic Processes" in Augustin et al.'s Introduction to Imprecise Probabilities, there is a definition of an upper transition operator as $$\overline T=I_{\cal X} \max$$ $\overline T$ is an upper prevision (upper expectation) operator, where f...

In Miranda and de Cooman's chapter 3, "Structural judgements", in Augustin et al.'s Introduction to Imprecise Probability, example 3.4 on p. 65 shows that independence in the selection (type-2 independence) does not imply strong independence (type-3 independence) for lower previsions. One of the...

Let $X$ be, for simplicity, a finite set (with the discrete topology). Denote with $M(X)$ the set of probability measures on $X$ endowed with the weak topology. For $\mu\in M(X)$ and a (necessarily measurable) function $f:X\rightarrow[-1,1]$ denote with $E_{\mu}(f)$ the expected value of $f$. F...