  This paper resolves affirmatively Koplienko's conjecture of 1984 on existence
of higher order spectral shift measures. Moreover, the paper establishes
absolute continuity of these measures and, thus, existence of the higher order
spectral shift functions $\eta_n$. We show the higher order spectral shift
function is a $L^1$-function and prove an estimate on its $L^1$-norm.
  Existence and summability of $\eta_1$ and $\eta_2$ were established by Krein
in 1953 and Koplienko in 1984, respectively, whereas for $n > 2$ the problem
was unresolved. Our method is derived from [arXiv:0904.4095]; it also applies
to the general semi-finite von Neumann algebra setting of the perturbation
theory.
