Let $1< p <+\infty$ and let $v$ be a non-decreasing weight on the interval $(0,+\infty)$. We prove that if the averaging operator $(Af)(x):=\frac{1}{x}\int_0^xf(t)dt$, $x\in(0,+\infty)$, is bounded on the weighted Lebesgue space $L^p((0,+\infty);v)$, then there exist $\varepsilon_0\in(0,p-1)$ such that the operator $A$ is also bounded on the space $L^{p-\varepsilon}((0,+\infty);v(x)^{1+\delta}x^{\gamma})$ for all $\varepsilon$, $\delta$, $\gamma\in[0,\varepsilon_0)$. Conversely, assuming that the operator $A$ is bounded on the space $L^{p-\varepsilon}((0,+\infty);v(x)^{1+\delta}x^{\gamma})$ for some $\varepsilon\in[0,p-1)$, $\delta\geq 0$ and $\gamma\geq 0$, we prove that the operator $A$ is bounded on the space $L^p((0,+\infty);v)$. Results have been obtained in collaboration with my colleague Ji\v{r}\'{\i} R\'akosn\'{\i}k.