I want to define a fucntional on the space of random variables continuous with respect to small relative perturbations of the underlying probability measure. To explain this, consider probability space with 2 elements, and denote $P(a)$ probability measure that assigns $a$ to the first one (whence $1-a$ to the second one). Then, for me, $P(0.19)$ is a good approximation of $P(0.2)$ (relative error 5%), but $P(0)$ is very far from $P(0.01)$ (relative error 100%). More formally, I want a convergence of probability measures such that $P(a+e)$ converges to $P(a)$ with $e->0$ if $a>0$ (and $1>a$), but $P(e)$ does NOT converge to $P(0)$.
Next, let $X$ be random variable and $F_X$ be its distribution function. I want to define a functional $G(F_X)$ continuous with respect to small relative perturbations of the corresponding probability space. For example, on the probability space above, all possible distributions are three-parameter family $(a,b,c)$ (r.v. $X$ is $b$ with probability $a$ and $c$ with probability $1-a$), so $G(F_X)=G(a,b,c)$. Let $G(a,b,c)=1$ if $a=0$ or $a=1$ (that is, for constant r.v.s), and $0$ otherwise. Then $G$ should be continuous (for the same reason why $P(e)$ does NOT converge to $P(0)$).
Questions
1) Does there exist such a convergence of probability measures? Something similar to convergence with respect to the total variation metric, but with ratio instead of difference?
2) Does there exist a corresponding convergence of random variables, such that my continuity would naturally arise as continuity with respect to this convergence?
