Our goal is to prove weak type \((1,1)\) boundedness for an operator \(T_A\) which is given as an average of a family of operators \(\{T_j\}_j\) satisfying certain estimates in the context of weighted Lebesgue spaces. In particular, we shall prove that, if there exists \(\alpha >0\) , \(s>0\) and \(C>0\) such that, for every weight v in the Muckenhoupt class \(A_p\) and every measurable set E, \(\displaystyle \sup _j \Vert T_j\chi _E\Vert _{L^{p, \infty }(v)} \le \frac C{(p-1)^s} \Vert v\Vert _{A_p}^\alpha v(E). \) and, for some \(u_0\in A_1\) fixed, \(\displaystyle \sup _j \Vert T_j\chi _E\Vert _{L^{1, \infty }(u_0)} \le C_{u_0} u_0(E), \) then, for every \(\beta >0\) , there exists a constant \(C_\beta >0\) , so that \(\displaystyle \begin{aligned} \sup _{y>0} \frac {y}{\Big (1+\log ^+(1+ \log ^+ \frac 1y) \Big )^\beta }\int _{ \{ |T_A \chi _E|>y \}} u_0(x) dx \le C_\beta \,u_0(E). \end{aligned} \) Our main technique is inspired on the theory of analytic families of operators and it is closely related to the Rubio de Francia’s extrapolation theory.