  We introduce a new decomposition technique for random variables that maps a
generic instance of the prophet inequalities problem to a new instance where
all but a constant number of variables have a tractable structure that we refer
to as $(\varepsilon, \delta)$-smallness. Using this technique, we make progress
on several outstanding problems in the area:
  - We show that, even in the case of non-identical distributions, it is
possible to achieve (arbitrarily close to) the optimal approximation ratio of
$\beta \approx 0.745$ as long as we are allowed to remove a small constant
number of distributions.
  - We show that for frequent instances of prophet inequalities (where each
distribution reoccurs some number of times), it is possible to achieve the
optimal approximation ratio of $\beta$ (improving over the previous best-known
bound of $0.738$).
  - We give a new, simpler proof of Kertz's optimal approximation guarantee of
$\beta \approx 0.745$ for prophet inequalities with i.i.d. distributions. The
proof is primal-dual and simultaneously produces upper and lower bounds.
  - Using this decomposition in combination with a novel convex programming
formulation, we construct the first Efficient PTAS for the Optimal Ordering
problem.
