When do 2 distribution on the real line is close?

Question

$f,g$ are two distribution on the real line. When are they considered to be close? $\int_\mathbb{R} |f-g|dx$ (or $L_2$ difference) are two ways but anybody is using $\int_\mathbb{R} |f-g|{fg}dx/\int_\mathbb{R}fgdx$ (conditional expectation of |f-g|), (or $\int_\mathbb{R} |f-g|{f}dx/\int_\mathbb{R}fdx$, if $g$ is approximating the true $f$)? Makes much more sense to me, since if an $x$ value has small probability w.r.t. fg, why would I worry about f not being close to g?