practical purposes, members of $\Pi$ can be viewed as singletons.

By default, each part of the common refinement contains a single outcome.

Observation spaces {#sec:space}
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An *observation space* is a triple $(\Omega,{\ensuremath{\mathrm{CO}}},{\mathcal P})$, such that

-   $(\Omega,{\ensuremath{\mathrm{CO}}})$ is an observation frame,

-   ${\mathcal P}$ is a real-valued function on ${\text{Ob}}=\bigcup{\ensuremath{\mathrm{CO}}}$ satisfying the following versions of the probability laws PL1–PL3.

    -   ${\mathcal P}(\Omega)=1$.

    -   If $(e_1,e_2)\in{\ensuremath{\mathrm{CO}}}$ and $e_1\cap e_2=\emptyset$ then ${\mathcal P}(e_1 + e_2) = {\mathcal P}(e_1) + {\mathcal P}(e_2)$.

    -   ${\mathcal P}(e)\ge0$ for all $e\in{\text{Ob}}$. $\triangleleft$

Since each observable event belongs to some ensemble, it suffices to define ${\mathcal P}$ on every ensemble (and ensure that every event gets the same probability in every ensemble that contains it). Since any ensemble $E$ is a Boolean algebra of sets, the probability distribution on $E$ is determined by the probabilities assigned to the parts of the ensemble partition $\Pi_E$.

Given an observation space $(\Omega,{\ensuremath{\mathrm{CO}}},{\mathcal P})$, do the following.

1.  Decide whether there is a traditional probability distribution that extends ${\mathcal P}$ from observable events to all events.

2.  If such a traditional extension exists then find one.

3.  Otherwise decide whether there is a signed probability distribution that extends ${\mathcal P}$ 