Convergence of integral of multiplication of two positive functions I have two functions $f, g:\mathbb{R}\rightarrow \mathbb{R}_{\ge0}$, that are continuous. I know that $\int\limits_{-\infty}^\infty f(s) \, ds=C_1&lt;\infty$, and $g(s)\le C_2$, with $C_1&gt; 0$ and $C_2\ge0$, for all $s\in\mathbb{R}$. 

For me, it is true that
$\int\limits_{-\infty}^\infty f(s)g(s) \, ds\le C_3&lt;\infty$, with $C_3\ge0$. This statement would come from the fact that $f(s)g(s)\le C_2f(s)$, and then integrating on both sides. 

This seems to be trivial, but I don't know if I should have additional conditions on the functions $f$ and $g$ so the previous statement is true. Any insights on this will be appreciated.
g(s) could be equal to a constant value per your statement then the integral would not be bounded.
To be true g(s) integral must be bounded also.
g(s) could be equal to a constant value per your statement then the integral would not be bounded.
To be true g(s) integral must be bounded also.