I'm working my way through Stein and Shakarchi's Real Analysis, and I'm having some trouble figuring this exercise out.
Given the function $K_\delta$ that satisfies the normal approximation to the identity properties
$|K_\delta(x)| \leq A\delta^{−d}$ for all $\delta > 0$
$|K_\delta(x)| \leq A\delta/|x|^{d+1}$ for all $\delta > 0$
but
$\int_{-\infty}^\infty{K_\delta(x)dx}= 0$ for all $\delta > 0$
prove that if $f$ is integrable on $R^d$ then $(f*K_\delta)(x) \to 0$ as $\delta \to 0$ for a.e $x$.
Is it enough to use $\int_{R^d}(f*K_\delta)(x)=\int_{R^d}f\int_{R^d}K_\delta$? The domain of integration is confusing me - integrating over $R^d$ the same as integrating over $-\infty$ to $\infty$ in this case, correct? Then $\int_{R^d}(f*K_\delta)(x)=0$. Am I missing something?
Thanks for the help.