work with finite spaces.

By a *signed probability space* $S$ we mean a pair $(\Omega,{\mathcal P})$ where $\Omega$ is a nonempty set and ${\mathcal P}$ is a real-valued function on $2^\Omega$ such that the following probability laws hold.

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

2.  If $e_1,e_2\subseteq\Omega$ and $e_1\cap e_2=\emptyset$ then ${\mathcal P}(e_1\cup e_2) = {\mathcal P}(e_1) + {\mathcal P}(e_2)$.

If, in addition, we have

1.  ${\mathcal P}(e)\ge0$ for all $e\subseteq\Omega$,

then ${\mathcal P}$ and $S$ are *traditional*. $\triangleleft$

Here $\Omega$ is the *sample space*, and its elements are *sample points* or *outcomes*. Subsets of $\Omega$ are *events*. ${\mathcal P}$ is a *signed probability distribution*. ${\mathcal P}(e)$, even if it is negative, is called the *probability* of $e$. For brevity, when $\omega\in\Omega$, we write ${\mathcal P}(\omega)$ to mean ${\mathcal P}(\{\omega\})$.

The complement $\Omega-e$ of an event $e$ will be denoted $\bar e$. The collection of all subsets of a set $s$ will be denoted $2^s$. Disjoint union of sets $s_1,s_2$ will be denoted $s_1+s_2$. If $S$ is a set of sets then $\displaystyle \bigcup S = \bigcup_{s\in S} s$.

In the example above, the sample space $\Omega$ consists of the four binary strings $00, 01, 10, 11$, and the probability distribution ${\mathcal P}$ 