  In science, the most widespread statistical quantities are perhaps
$p$-values. A typical advice is to reject the null hypothesis $H_0$ if the
corresponding p-value is sufficiently small (usually smaller than 0.05). Many
criticisms regarding p-values have arisen in the scientific literature. The
main issue is that in general optimal p-values (based on likelihood ratio
statistics) are not measures of evidence over the parameter space $\Theta$.
Here, we propose an \emph{objective} measure of evidence for very general null
hypotheses that satisfies logical requirements (i.e., operations on the subsets
of $\Theta$) that are not met by p-values (e.g., it is a possibility measure).
We study the proposed measure in the light of the abstract belief calculus
formalism and we conclude that it can be used to establish objective states of
belief on the subsets of $\Theta$. Based on its properties, we strongly
recommend this measure as an additional summary of significance tests. At the
end of the paper we give a short listing of possible open problems.
