  We study a class of measure-theoretic objects that we call cubic couplings,
on which there is a common generalization of the Gowers norms and the Host-Kra
seminorms. Our main result yields a complete structural description of cubic
couplings, using nilspaces. We give three applications. Firstly, we describe
the characteristic factors of Host-Kra type seminorms for measure-preserving
actions of countable nilpotent groups. This yields an extension of the
structure theorem of Host and Kra. Secondly, we characterize sequences of
random variables with a property that we call cubic exchangeability. These are
sequences indexed by the infinite discrete cube, such that for every integer
$k\geq 0$ the joint distribution's marginals on affine subcubes of dimension
$k$ are all equal. In particular, our result gives a description, in terms of
compact nilspaces, of a related exchangeability property considered by Austin,
inspired by a problem of Aldous. Finally, using nilspaces we obtain limit
objects for sequences of functions on compact abelian groups (more generally on
compact nilspaces) such that the densities of certain patterns in these
functions converge. The paper thus proposes a measure-theoretic framework on
which the area of higher-order Fourier analysis can be based, and which yields
new applications of this area in a unified way in ergodic theory and arithmetic
combinatorics.
