I have been reading some game theory and I have come across the following belief-based definitions. As is recommended, it is instructive to prove the equivalence between definition 3 and the usual 'rival strategy profile' based definition of strictly dominant strategy (similarly definition 6 and the usual 'rival strategy profile' - based weakly dominant strategy). I don't have a problem in doing that. But the source I am reading has a pedagogical gap it seems as it doesn't provide definition 4. Can somebody provide that?
Note: Script U denotes the expected utility over all $s_{-i} \in S_{-i}$
Definition 1: strictly dominated strategy with belief $\mu_{i}$
A strategy $t_i \in S_i$, for player $i$, is strictly dominated by strategy $s_i \in S_i $ with respect to belief $\mu_i \in \Delta (S_{-i})$ if
$\mathscr{U}_i(s_i,\mu_i) > \mathscr{U}(t_i, \mu_i)$
We write, $s_i \succ_{\mu_i} t_i$
Definition 2: Strictly dominated strategy
A strategy $t_i \in S_i$, for player $i$, is strictly dominated by strategy $s_i \in S_i $ if
$\mathscr{U}_i(s_i,\mu_i) > \mathscr{U}_i(t_i, \mu_i) \space \forall \mu_i \in \Delta (S_{-i})$
Definition 3: Strictly dominant strategy
A strategy $s_i \in S_i$, for player $i$, is a strictly dominant strategy if it strictly dominates strategy $t_i \in S_i $ for all $t_i \in S_i $
Definition 4 Weakly dominated strategy with belief $mu_i$
I think this can't be defined? Please confirm.
Definition 5 Weakly dominated strategy
A strategy $t_i \in S_i$, for player $i$, is weakly dominated by strategy $s_i \in S_i $ if
$\mathscr{U}_i(s_i,\mu_i) \ge \mathscr{U}_i(t_i, \mu_i) \space \forall \mu_i \in \Delta (S_{-i})$
with strict inequality for at least one $ \hat{\mu_i} \in \Delta (S_{-i}) $
Definition 6 Weakly Dominant Strategy
A strategy $s_i \in S_i$, for player $i$, is weakly dominant strategy if it weakly dominates strategy $t_i \in S_i $ for all $t_i \in S_i$
