I have read that a sufficient condition for a family of approximations of the identity $K_{\epsilon}={\epsilon}^{-n}K(x/{\epsilon})$, with K integrable and $\int K =1$ for wich the convolution converges a.e. pointwise to f,
$K_{\epsilon}\ast f(x) \xrightarrow[\varepsilon \to 0]{} f(x)$ $\quad$ a. e. x,
is that
$\ell(x)=sup\{K(y):|y|\geq|x|\}$ , the minimal non decreasing radial majorant of K, must be integrable.
But this tell us that maybe the merely integrability of K is not a sufficient condition for the convergence. In fact, it's actually not, but I don't know how to find a counterexample! Please can somebody help me?