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In this paper, the stability analysis problem for a class of switched positive linear systems (SPLSs) with average dwell time switching is investigated. A multiple linear copositive Lyapunov function (MLCLF) is first introduced, by which the sufficient stability criteria in terms of a set of linear matrix inequalities, are given for the underlying systems in both continuous-time and discrete-time contexts. The stability results for the SPLSs under arbitrary switching, which have been previously studied in the literature, can be easily obtained by reducing MLCLF to the common linear copositive Lyapunov function used for the system under arbitrary switching those systems. Finally, a numerical example is given to show the effectiveness and advantages of the proposed techniques.