We say that a function \(\delta\) (x) is a control function for an almost everywhere convergence of \(f_ n(x)\) to f(x) on [0,1], if for every \(\epsilon >0\) there exists an integer n(\(\epsilon)\) such that \(| f_ n(x)-f(x)| 0\), \(\gamma =\min (1,k/2)\) and \(\{\beta >1-\alpha /k\), \(01-\gamma /k\}\). If \(\sigma_ n^{\beta}(x)\) converges to f(x) a.e. on [0,1] and it has an \(L^ p\)- integrable control function for some \(p>0\), then \[ \lim_{n\to \infty}(1/A_ n^{\alpha})\sum^{n}_{\nu =0}A^{\alpha -1}_{n- \nu}| \sigma_{\nu}^{\beta -1}(x)-f(x)|^ k=0 \] a.e. on [0,1], and the above convergence has an \(L^ q\)-integrable control function, where \(q=\min (2,p)\).