  In this paper we prove the existence of an invariant measure for the cubic
NLS $$i\partial_t u + \bigtriangleup u - |u|^2 u = 0$$ on the real line in the
sense that we prove the existence of a measure $\rho$ supported by
non-localised functions such that there exists random variables $X(t)$ whose
laws are $\rho$ (thus independent of $t$) and such that $t\mapsto X(t)$ is a
solution to the cubic NLS. Our strategy for the proof is inspired by
\cite{burqtzv} and relies on the application of Prokhorov and Skorokhod
Theorems to a sequence of measures which are invariant under some approximating
flows, as we proved in our previous \cite{lastbaby}. However, the work by
Bourgain, \cite{B00} provides a stronger result than this one, as it gives
almost sure strong solutions for the cubic NLS and the invariance of the
measure can be deduced from it.
