  We consider the set of all initial states within a microcanonical energy
shell of an isolated many-body quantum system, which exhibit the same,
arbitrary but fixed non-equilibrium expectation value for some given observable
$A$. On condition that this set is not too small, it is shown by means of a
dynamical typicality approach that most such initial states exhibit
thermalization if and only if $A$ satisfies the so-called weak eigenstate
thermalization hypothesis (wETH). Here, thermalization means that the
expectation value of $A$ spends most of its time close to the microcanonical
value after initial transients have died out. The wETH means that, within the
energy shell, most eigenstates of the pertinent system Hamiltonian exhibit very
similar expectation values of $A$.
