The convergence of a net in two integrals

Question

Is there a net of complex functions $\{f_\alpha:[0,+\infty)\to C\}_{\alpha\in\Gamma}$ such that for all $\alpha\in\Gamma$, $$\int_0^\infty|f_\alpha(t)|e^{-t}dt<\infty,$$ and $$\lim_\alpha\int_0^\infty f_\alpha(t)e^{-t}dt=1,$$ but $$\lim_\alpha\int_0^\infty f_\alpha(t)e^{-2t}dt=0$$?

Answer 1

Maybe I'm missing something, but what if you let (for all $\alpha$) $$ f_\alpha(t)=4\sin t-2\cos t? $$ It will certainly satisfy your first bound, $$ \int_0^{+\infty}|f_\alpha(t)|e^{-t}\,dt\leq 6\int_0^{+\infty} e^{-t}\,dt=6. $$ Moreover, you will have (taking Laplace transforms) $$ \int_0^{+\infty} f_\alpha(t)e^{-st}\,dt=\frac{4-2s}{1+s^2}. $$ So in particular $$ \int_0^{+\infty} f_\alpha(t)e^{-t}\,dt=1 $$ and $$ \int_0^{+\infty} f_\alpha(t)e^{-2t}\,dt=0. $$