Let $f,g$ be functions in $C_b\left(\left[\varepsilon,\frac{1}{2}\right]\right)$, $\forall \; \varepsilon > 0$, or equivalently $f,g \in C\left(0,\frac{1}{2}\right]$.
We have $f(p) \overset{p \to 0}{\sim}g(p) $, meaning that $\limsup_{p \to 0}\left\lvert\frac{f(p)}{g(p)}\right\rvert < \infty$ and $\limsup_{p \to 0}\left\lvert\frac{g(p)}{f(p)}\right\rvert < \infty$.
My question is, if it holds that \begin{align*} \int_p^{\frac{1}{2}}f(x)dx \overset{p \to 0}{\sim}\int_p^{\frac{1}{2}}g(x)dx \; \; \; ? \end{align*}
I would be happy for any ideas.
The integrals are affected by the behavior of f, g everywhere and not just near zero, so I'd say the answer is no.