introduced by Smets [25], we can also define the belief function only with:

$$({\mathfrak{I}})$$
$$(4)$$
$$(5)$$
3
$$\sum_{A\in2^{\Theta}}m(A)=1.$$

m(A**) = 1**. (3)
and thus we can have m(∅) > 0, working with the open-world assumption. In order to change an open world to a closed world, we can always add one extra closure element in the open discriminant space Θ. In the following, we assume that we always work within a closed-world Θ.

The (quantitative) basic belief assignment (bba) m(.) can also be defined similarly in the DSmT framework by working on hyper-power set DΘ instead on classical power-set 2 Θ as within DST. More generally for taking into account some integrity constraints on (closed-world) Θ (if any), m(.) can be defined on GΘ as:

$$m(\varnothing)=0,$$
m(∅**) = 0**, (4)
$$\sum_{A\in G^{\Theta}}m(A)=1.$$

m(A**) = 1**. (5)
The conditions (1)-(5) give a large panel of definitions of the belief functions, which is one of the difficulties of the theories.

From any basic belief assignments m(.), other belief functions can be defined such as the credibility Bel(.) and the plausibility Pl(.) [17], [18] which are in one-to-one correspondence with m(.).

After combining several bba's provided by several sources of evidence into a single one with some chosen fusion rule
(see next section), one usually has also to make a final decision to select the "best" hypothesis representing the unknown truth for the problem under consideration. Several approaches are generally adopted for decision-making from belief functions m(.), Bel(.) or Pl(.). The maximum of the credibility function Bel(.) is known to provide a pessimistic decision, while the maximum of the plausibility function Pl(.) is often considered as too optimistic. A common solution for decision-making in these frameworks is to use the *pignistic probability* denoted **BetP(**X) [25] which offers a good compromise between the max of Bel(.) and the max of Pl(.). The pignistic probability in DST framework is given for all X ∈ 2 Θ, with X 6= ∅ by:

of $m(0)$ and $m(0)$. The $p$-th population probability of $m(0)$ is given by  $$\text{BetP}(X)=\sum_{Y\in\mathbb{Z}^{n},\,Y\neq\emptyset}\frac{|X\cap Y|}{|Y|}\frac{m(Y)}{1-m(0)},\tag{6}$$  for $m(0)\neq1$. The $p$-th population probability can also be defined in DSmT framework as well (see Chapter 7 of [18] for details).  
$$(6)$$
When we can quantify/estimate the reliability of each source of evidence, we can weaken the basic belief assignment before the combination by the classical discounting procedure [17]:

$m^{\prime}(X)=\alpha m(X)$, $\forall X\in2^{\Theta}\setminus\{\Theta\}$  $m^{\prime}(\Theta)=\alpha m(\Theta)+1-\alpha$.  

$$\left(T\right)$$
α ∈ [0, 1] is the discounting factor of the source of evidence B that is in this case the reliability of the source of evidence B, eventually as a function of X ∈ 2 Θ. Same procedure can be applied for bba's defined on GΘ in DSmT framework.