  In this manuscript, we appeal to Potential Theory to provide a sufficient
condition for existence of distributional solutions to fractional elliptic
problems with non-linear first-order terms and measure data $\omega$: $$
\left\{ \begin{array}{rcll} (-\Delta)^su&=&|\nabla u|^q + \omega \quad \text{in
}\mathbb{R}^n,\, \,\,s \in (1/2, 1)\\u & > &0 \quad \text{in }
\mathbb{R}^{n}\\\lim_{|x|\to \infty}u(x) & =& 0, \end{array} \right. $$under
suitable assumptions on $q$ and $\omega$. Roughly speaking, the condition for
exis\-tence states that if the measure data is locally controlled by the Riesz
fractional capacity, then there is a global solution for the equation. We also
show that if a positive solution exists, necessarily the measure $\omega$ will
be absolutely continuous with respect to the associated Riesz capacity, which
gives a partial reciprocal of the main result of this work. Finally, estimates
of $u$ in terms of $\omega$ are also given in different function spaces.
