In this paper, we address the positive filtering problem for positive continuous-time systems under the L-1-induced performance. A pair of positive filters with error-bounding feature is proposed to estimate the output of positive systems. A novel characterisation is first obtained to ensure that the filtering error system is asymptotically stable with a prescribed L-1-induced performance. Then, necessary and sufficient conditions for the existence of required filters are presented, and the obtained results are expressed in terms of linear programming problems, which can be easily checked by standard software. Finally, a numerical example is given to illustrate the effectiveness of the proposed design procedures.