  We study the amount of randomness needed for an input process to approximate
a given output distribution of a channel in the $E_{\gamma}$ distance. A
general one-shot achievability bound for the precision of such an approximation
is developed. In the i.i.d.~setting where $\gamma=\exp(nE)$, a (nonnegative)
randomness rate above $\inf_{Q_{\sf U}: D(Q_{\sf X}||\pi_{\sf X})\le E}
\{D(Q_{\sf X}||\pi_{\sf X})+I(Q_{\sf U},Q_{\sf X|U})-E\}$ is necessary and
sufficient to asymptotically approximate the output distribution $\pi_{\sf
X}^{\otimes n}$ using the channel $Q_{\sf X|U}^{\otimes n}$, where $Q_{\sf
U}\to Q_{\sf X|U}\to Q_{\sf X}$. The new resolvability result is then used to
derive a one-shot upper bound on the error probability in the rate distortion
problem, and a lower bound on the size of the eavesdropper list to include the
actual message in the wiretap channel problem. Both bounds are asymptotically
tight in i.i.d.~settings.
