Is there a commonly used name for the property of a null hypothesis for being "weakest possible" ?
There is in general several possibilities to write $H0$ for a specific test. All possible $H0$ ensure that $H0 \implies \forall \alpha, P(p_{value}\leq\alpha)\leq \alpha$. Some of them ensure that $H0 \iff \forall \alpha, P(p_{value}\leq\alpha)\leq \alpha$ and those $H0$ are then weakest possible but I do not find this name on litterature.
I guess this is more or less equivalent to the property (under possible assumption) :
$H1=\bar{H0} \implies \forall \alpha\in]0,1], \forall \beta\in]0,1], \exists K\in \mathbb{N}, n>K=>P(p_{value}<\alpha)>1-\beta$
(In other words when n tends towards $+\infty$, not rejecting $H0$ is equivalent to accept it.)
For most current test $H0$ there is an easily writtable $H0$ which is weakest possible. Let's take as example the Mann Whitney rank-sum test for which there is two commonly used defintion of hyopthesis (without counting the erroneous ones) :
Preiliminary hyopthesis : "the two distributions of X and Y are the same, except of a possible shift" ; $H0$ is then : "The shift is $0$".
This $H0$ is well "weakest possible".There is no preliminary assumption and $H0$ is "X and Y follow the same distribution". This assumption is not weakest possible. As example with N=2, $P(X=0)=1, P(Y=1)=.5, P(Y=-1)=-.5$ , X and Y do not follow the same distribution but $\forall \alpha, P(p_{value}\leq\alpha)\leq\alpha$. Another example which do not respect the second property : X and Y follow two gaussian distributions with the same average and different variance. They are not following the same distribution but $P(p_{value}<\alpha)$ do not tends towards 1. (Note however on this case $\exists \alpha,P(p_{value}<\alpha)>\alpha$ )
I guess this may be a commonly discussed topic but I do not have the right keywords to find references.