We give a necessary and sufficient condition for a positive measurable kernel ${\bold C}(\cdot)$ to satisfy $$ \int_1^xf(t){\bold C}(t)dt\sim f(x)\int_1^x\bold C(t)dt\qquad(x\to\infty) $$ whenever $f(\cdot)$ is from the class of Karamata's regularly varying functions.