  We derive an upper bound on the difference between the long-time average and
the microcanonical ensemble average of observables in isolated quantum systems.
We propose, numerically verify, and analytically support a new hypothesis,
eigenstate randomization hypothesis (ERH), which implies that in the energy
eigenbasis the diagonal elements of observables fluctuate randomly. We show
that ERH includes eigenstate thermalization hypothesis (ETH) and makes the
aforementioned bound vanishingly small. Moreover, ERH is applicable to
integrable systems for which ETH breaks down. We argue that the range of the
validity of ERH determines that of the microcanonical description.
