For a given real $a>0$, I'm looking for a measurable function $g : [0,a] \rightarrow \mathbb{R}^+ \in L^1([0,a])$ (i.e $g$ is a density for a random variable valued in $[0,a]$) such that :
$\int_0^a |\ln x| g(x) \ dx $ is a divergent integrale.
$\int_0^a |\ln y| g(y) \left( \int_0^y g(x)\ dx \right) \ dy$ is a convergent integrale.
I actually have no idea if such a function exists, do you have any thoughts on this ?
I'm gonna choose $a<1$ to avoid sign issue on the log. The function :
$$g : x \mapsto \frac{1}{x |\ln(x)|^2}$$
should definitely work using Bertrand integrale convergence. To verify the second hypothesis we just have to note that :
$$\int_0^y g(x) dx = - \frac{1}{\ln(y) }$$ for $y < a < 1$.