A real or a complex valued sequence \((s_ k)\) is strongly convergent to a number t, of index \(\lambda >0\), if \[ \frac{1}{n+1}\sum^{n}_{k=0}| (k+1)(s_ k-t)-k(s_{k-1}- t)|^{\lambda}=o(1). \] Here \(s_{-1}=0\). We write \(s_ k\to t[I]_{\lambda}\). Theorem 1. If the coefficients of a trigonometric series (1) \(a_ o| 2+\sum^{\infty}_{k=1}(a_ k\quad \cos kx+b_ k\quad \sin kx)\) satisfy (2) \(\frac{1}{n+1}\sum^{n}_{k=0}k^{\lambda}| a_ k|^{\lambda}=o(1)\) for some \(\lambda >1\), then (1) is a Fourier series of a function f which belongs to \(L^ p\) for each \(p\geq 1\) and \(a_ n\to f[I]_{\lambda}\) at every Lebesgue point of f. We now state Theorem 3. There are trigonometric series that are strongly convergent almost everyhwere to a function which is not in \(L^ p\) for \(p>3/2\).