  Under unitary time evolution, expectation values of physically reasonable
observables often evolve towards the predictions of equilibrium statistical
mechanics. The eigenstate thermalization hypothesis (ETH) states that this is
also true already for individual energy eigenstates. Here we aim at elucidating
the emergence of ETH for observables that can realistically be measured due to
their high degeneracy, such as local, extensive or macroscopic observables. We
bisect this problem into two parts, a condition on the relative overlaps and
one on the relative phases between the eigenbases of the observable and
Hamiltonian.
