$$I=\int_{-\infty}^{\infty} \frac{\log|\zeta(\frac{1}{2}+it)|}{\frac{1}{4}+t^2}dt$$ where $\zeta$ is the Riemann zeta function. It is known by Balazard Saias and Yor Paper that I is integrable and $0\leq I<\infty$.
Since by Schwarz Reflection principle, integrand is an even function, so, $$I= 2\int_{0}^{\infty} \frac{\log|\zeta(\frac{1}{2}+it)|}{\frac{1}{4}+t^2}dt $$ Let, $0<t_1<t_2<t_3<...<t_n<t_{n+1}<...$ be infinitely many zeros of $\zeta(s)$ on the critical line $\Re(s)=1/2$. Define ,$ \ t_0=0$. $$I= 2 \ \sum_{n=0}^{\infty}\int_{t_n}^{t_{n+1}} \frac{\log|\zeta(\frac{1}{2}+it)|}{\frac{1}{4}+t^2}dt $$ Let, $\epsilon>0$ be arbitrarily small. Define, $$I_{\epsilon}=2\sum_{n=0}^{\infty}\int_{t_n+\frac{\epsilon}{4}}^{t_{n+1}-\frac{\epsilon}{4}} \frac{\log|\zeta(\frac{1}{2}+it)|}{\frac{1}{4}+t^2}dt $$ Note that $|\zeta(\frac{1}{2}+it)|$ is non zero on $[t_n+\frac{\epsilon}{4},t_{n+1}-\frac{\epsilon}{4}]$ $\forall n\geq 0$, so $\log|\zeta(\frac{1}{2}+it)|$ is well defined on $[t_n+\frac{\epsilon}{4},t_{n+1}-\frac{\epsilon}{4}]$ $\forall n\geq 0$
Question 1 Prove that $$2\sum_{n=0}^{\infty}\lim_{\epsilon\to 0^{+}}\int_{t_n+\frac{\epsilon}{4}}^{t_{n+1}-\frac{\epsilon}{4}} \frac{\log|\zeta(\frac{1}{2}+it)|}{\frac{1}{4}+t^2}dt =I $$ Question 2 Prove that $$\lim_{\epsilon \to 0^{+}} I_{\epsilon}=I$$
Do I need Dominated Convergence Theorem?..
As there were many comments and the situation seems to still be confused for you, I will address some issues in one post.
(edited later per comments below); the crux of the matter here is that the statement (existing in the question version as of March 8, screenshot available upon request):
"Note that $|\zeta(\frac{1}{2}+it)|$ is non zero on $[t_n+\frac{\epsilon}{4},t_{n+1}-\frac{\epsilon}{4}]$ $\forall n\geq 0$ so $\log|\zeta(\frac{1}{2}+it)|$ is well defined etc" is incorrect.
Both questions $1,2$ (as stated now) are incorrect as formulated for the simple fact that given $\epsilon >0$ fixed, (at least and if current conjectures are true, actually all for some $n(\epsilon)$ on) some of the intervals $[t_n+\frac{\epsilon}{4},t_{n+1}-\frac{\epsilon}{4}]$ are ill-defined in the sense that the left end is higher than the right end, while even if we reverse them, infinitely many such intervals will eventually still contain other RZ roots, so there is no way to define a union of intervals as you want to avoid roots of RZ.
(for once $\liminf (t_{n+1}-t_n) =0$ showing first assertion above, and for another if we could have a union of intervals whether $[t_n+\frac{\epsilon}{4},t_{n+1}-\frac{\epsilon}{4}]$ or $[t_{n+1}-\frac{\epsilon}{4}, t_n+\frac{\epsilon}{4}]$ depending which end is higher, avoiding roots it would follow that roots are spaced at least $\epsilon/4$ apart and that trivially gives a contradiction with Selberg's theorem that there at least $cT\log T$ roots on the critical line with height up to $T$).
This being said I truly have no clue about the purpose of these questions since if one wants to prove that $I$ exists is finite and non-negative, there are various rigorous ways of doing it - the Balazard et others paper gives one based on the theory of Hardy spaces, another way can be given by constructing a domain where $\log \zeta$ is analytic by excluding horizontal segments ending at zeroes and using Cauchy to move the path of integration to infinity etc. Those methods give the full expression of $I$ in terms of a sum of non-trivial zeroes with real part greater than $1/2$ which can be easily estimated above by known results to a fairly small number, so $0 \le I < \delta$ with RH true iff $I=0$.
If one just wants to show that $I$ exists and is finite, that has nothing really to do with the theory of RZ beyond trivial estimates in the strip of the type $|\zeta(s)| \le C|s|, \Re s \ge \sigma >0$, the local form of holomorphic functions at zeroes and Lebesgue integration theory as one can trivially show that $\frac{log|\zeta(\frac{1}{2}+it)|}{\frac{1}{4}+t^2}$ is integrable on $[M,N]$ for $M>N \ge 0$ and that $\lim_{M >N \to {\infty}}\int_{N}^{M} \frac{log|\zeta(\frac{1}{2}+it)|}{\frac{1}{4}+t^2}dt=0$ which gives that $\int_{0}^{\infty} \frac{log|\zeta(\frac{1}{2}+it)|}{\frac{1}{4}+t^2}dt$ exists and is finite; actually pretty much same proof and estimates show that $ \frac{log|\zeta(\frac{1}{2}+it)|}{\frac{1}{4}+t^2}$ is absolutely integrable on $[0, \infty)$ though of course one doesn't have a neat form of the integral of the absolute value anymore.