In this note, we consider the existence of positive solution to \begin{equation}\label{eq1.1} \left{ \arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u=u_+^p+\sigma\lambda,\quad & \rm{in}\quad\Omega,\[2mm] u=0,\quad & \rm{in}\quad \R^N\setminus\Omega, \end{array} \right. \end{equation} where $p>0$, $\sigma>0$, $\lambda\in\mathfrak{M}(\Omega)$ with $\mathfrak{M}(\Omega)$ the Radon measure space, $u_+(x)=\max{u(x),0}$ and $\Omega$ is an open, smooth domain of $\R^N\ (N\ge2)$. Here $(-\Delta)^\alpha $ is defined, for a regular function $u$, as follow \begin{equation}\label{eq 1} (-\Delta)^\alpha u(x)=(\alpha-1)\lim_{r\to0}\int_{\R^N\setminus B_r}\frac{u(x+y)-u(x)}{|y|^{N+2\alpha}}dy, \end{equation} where $\alpha\in(0,1)$, $B_r$ denotes the ball centered at origin with radius $r$ in $\R^N$. This definition is called \emph{in the principle value sense.}
