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Using methods from [9] we prove the almost everywhere convergence of the Cesàro means of Laguerre series associated with the system of Laguerre functions $L_n^a\left(x\right)={(n!/\Gamma (n+a+1))}^{1/2}{e}^{-x/2}{x}^{a/2}{L}_n^a\left(x\right)$, n = 0,1,2,..., a ≥ 0. The novel ingredient we add to our previous technique is the $A_p$ weights theory. We also take the opportunity to comment and slightly improve on our results from [9].