Let be $|*|:S\rightarrow [0, +\infty]$ the Lebesgue measure on the $\mathbb{R}$ set, $\{q_n:n\geq1\}=\mathbb{Q}$ and the functions: $$ \begin{equation} \gamma(t) = \begin{cases} \frac1{\sqrt{t}} & 0<|t|<1 \\ 0& |t|=0 \ \lor |t|>1 \end{cases} \end{equation} $$ $$f(t)=\sum_{n\geq1}\frac1{5^n}\gamma(t-q_n)$$ with $t \in \mathbb{R}$. Prove that: $$a) \ f(t)<+\infty\ almost\ everywhere \ in \ \mathbb{R} \ and \int_{-\infty}^{+\infty}f=1$$ $$b) \ f\ is \ bounded \ in \ every \ interval \ I \subset\ \mathbb{R} \ that \ \mathring I \neq \emptyset$$ $$c) \ Even \ if \ the \ f \ function \ is \ modified \ on \ a \ negligible \ set \ the \ (b) \ propery\ \ still\ holds$$
I managed to integrate $\int_\mathbb{R}\gamma(t)dt=4$ then I showed that, because $\mathbb{Q}$ is a negligible set: $$\int_\mathbb{R}f_n(t)dt=\int_\mathbb{R}\frac1{5^n}\gamma(t-q_n)dt=\int_\mathbb{R}\frac1{5^n}\gamma(s)ds=\frac4{5^n}$$ Then using the monotone convergence theorem I proved that $\int_{\mathbb{R}}f(t)dt=1$. I actually have no idea how to visualize and prove (a) and (b).