Summary: The Whittaker-Shannon-Kotel’nikov sampling theorem enables one to reconstruct signals \(f\) bandlimited to \([-\pi W,\pi W]\) from its sampled values \(f(k/W)\), \(k\in\mathbb{Z}\), in terms of \[ (S_Wf)(t)\equiv \sum^\infty_{k=-\infty}f\left(\frac kW\right)\text{sinc}(W_t-k)=f(t)\;(t\in\mathbb{R}). \] If \(f\) is continuous but not bandlimited, one normally considers \(\lim_{W\to\infty}(S_Wf)(t)\) in the supremum-norm, together with aliasing error estimates, expressed in terms of the modulus of continuity of \(f\) or its derivatives. Since in practice signals are however often discontinuous, this paper is concerned with the convergence of \(S_Wf\) to \(f\) in the \(L^p(\mathbb{R})\)-norm for \(1<p<\infty\), the classical modulus of continuity being replaced by the averaged modulus of smoothness \(\tau_r(f; W^{-1};M(\mathbb{R}))_p\). The major theorem enables one to sample any bounded signal \(f\) belonging to a certain subspace \(\Lambda^p\) of \(L^p(\mathbb{R})\), the jump discontinuities of which may even form a set of measure zero on \(\mathbb{R}\). A corollary gives the counterpart of the approximate sampling theorem, now in the \(L^p\)-norm. The averaged modulus, so far only studied for functions defined on a compact interval \([a,b]\), had first to be extended to functions defined on the whole real axis \(\mathbb{R}\). Basic tools are the de La Vallée Poussin means and a semi-discrete Hilbert transform.