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In a previous paper, we proved that the 1-d periodic fractional Schrödinger equation with cubic nonlinearity is locally well-posed in 
${{H}^{s}}$ for 
$s>1-\alpha /2$ and globally well-posed for 
$s>10\alpha -1/12$. In this paper we define an invariant probability measure 
$\mu$ on ${{H}^{s}}$ for 
$s<\alpha -1/2$, so that for any 
$\text{ }\!\!\varepsilon\!\!\text{ }>0$ there is a set 
$\Omega \subset {{H}^{s}}$ such that 
$\mu \left( {{\Omega }^{c}} \right)<\text{ }\!\!\varepsilon\!\!\text{ }$ and the equation is globally well-posed for initial data in 
$\Omega$. We see that this fills the gap between the local well-posedness and the global well-posedness range in an almost sure sense for 
$\frac{1-\alpha }{2}<\alpha -\frac{1}{2},i.e.,\alpha >\frac{2}{3}$ in an almost sure sense.
