  We study the problem of finding the value of $p$ and the summability index
$\delta$ for which the Bochner-Riesz operator $S_t^\delta f(x) :=
(1+t^{-2}\Delta)_+^\delta f(x)$ converges to $f(x)$ almost everywhere for all
$f\in L^p(\mathbb R^d)$ in the range $2\le p\le \infty$. Although the
convolution kernel $K_t^\delta$ of $S_t^\delta$ is not in $L^{p'}$ if
$\delta\le\delta(d,p):= \max(0,-1/2 + d\,|1/2-1/p|)$, $S_t^\delta f$ can be
defined for some $f\in L^p$ in such a case by interpreting the convolution
$K_t^\delta \ast f$ as an improper integral. Using this alternative definition,
we prove that the almost everywhere convergence holds only if $\delta \ge
\delta(d,p)$.
