  We deal with a global Calder\'on-Zygmund type estimate for elliptic obstacle
problems of $p$-Laplacian type with measure data. For this paper, we focus on
the singular case of growth exponent, i.e. $1<p \le 2-\frac{1}{n}$. In
addition, the emphasis of this paper is in obtaining the Lorentz bounds for the
gradient of solutions with the use of fractional maximal operators.
