Previously, I asked this question to the forum.
Now, a question in this regard is coming to my mind. Suppose, we don't have the integral; i.e. we have the expression:
$(\omega_{\epsilon}(t-t_{1}) - \omega_{\epsilon}(t-t_{2}))$ & multiplied by $f_{\epsilon}$ (i.e. $f * \omega_{\epsilon})$ . & we ask the same question.
i.e.: What will be the limit: $\lim _{\epsilon \to 0} (\omega_{\epsilon}(t-t_{1}) - \omega_{\epsilon}(t-t_{2})) f_{\epsilon} $ ??
I am asking this question just based on curiosity!! Because I know $f_{\epsilon} \to f$ a.e. as $\epsilon \to 0$ ; But what about the additional $\omega_{\epsilon}(t-t_{1}) - \omega_{\epsilon}(t-t_{2})$ ??