  We study positive solutions to the fractional Lane-Emden system
\begin{equation*} \tag{S}\label{S} \left\{ \begin{aligned} (-\Delta)^s u &=
v^p+\mu \quad &&\text{in } \Omega \\ (-\Delta)^s v &= u^q+\nu \quad &&\text{in
} \Omega\\ u = v &= 0 \quad &&\text{in } \Omega^c={\mathbb R}^N \setminus
\Omega, \end{aligned} \right. \end{equation*} where $\Omega$ is a $C^2$ bounded
domains in ${\mathbb R}^N$, $s\in(0,1)$, $N>2s$, $p>0$, $q>0$ and $\mu,\, \nu$
are positive measures in $\Omega$. We prove the existence of the minimal
positive solution of the above system under a smallness condition on the total
mass of $\mu$ and $\nu$. Furthermore, if $p,q \in (1,\frac{N+s}{N-s})$ and $0
\leq \mu,\, \nu\in L^r(\Omega)$ for some $r>\frac{N}{2s}$ then we show the
existence of at least two positive solutions of the above system. We also
discuss the regularity of the solutions.
