Power one sequential tests exist for weakly compact mathscr P against mathscr Pc

We study power-one sequential testing for an i.i.d. law on a Polish sample space. Given a nonempty composite null class Pcalsubseteqmathcal M1(X), we ask when there exists a level-α stopping rule that rejects almost surely under every alternative in a prescribed class QcalsubseteqPcalc. Our main sufficient condition is local weak lower semicontinuity and positivity of the information projection functional ( Phi_Pcal(Q):=infP∈PcalKL(Q|P). ) In particular, if Pcal is weakly compact, then for every α∈(0,1) there is a single level-α sequential test with power one against the entire complement Pcalc. The proof combines Csisz'ar's nonasymptotic Sanov bound for weakly closed convex empirical-measure sets with a Lindel"of countable-subcover argument. We also show that weak lower semicontinuity is sufficient but not necessary by giving examples where discontinuous finite-sample events separate alternatives that weak neighborhoods cannot detect. Finally, we construct an e-process that is asymptotically relatively growth-rate optimal under weak compactness. We verify the weak-lower-semicontinuity condition for weakly compact nulls, f-divergence balls, several integral probability metric balls, Wasserstein balls on proper spaces, and a number of non-weakly-compact semiparametric examples.