Suppose I want to define a space $\mathcal L^1(\mathbb R)$ the set of function s.t. $-\infty <\int_{\mathbb R}f<\infty $. And now I say that a sequence $(f_n)$ of element of $\mathcal L^1(\mathbb R)$ converge to a function $f\in \mathcal L^1$ if $$\lim_{n\to \infty }\int_{\mathbb R}f_n=\int_{\mathbb R}f.$$ Is there a way to find a topology from that ? What would be the open of this set ?
I know that that my topological space is not really interesting... and that it won't be hausdorff since for example $f_n=\boldsymbol 1_{[n,n+1]}$ will have a lot of different limits (for example it will converges to $f_a=\boldsymbol 1_{[a,a+1]}$ for all $a\in\mathbb R$, the we'll have at least an uncountable number of different limit). But I would be very interested if we can find the open of my new $\mathcal L^1(\mathbb R)$ space.