Let $X$ and $Y$ be two continuous random variables with PDF $f(x)$ and $g(y)$. Define $P_{e1} = \Pr[X \leq y \mid Y>y ]$, and $P_{e2} = \Pr[X > y \mid Y \leq y ]$. For any given constant $0 \leq \alpha \leq 1$, we define $P(f,g) = 1 - \alpha P_{e1} - ( 1- \alpha) P_{e2}$, respectively. Consider appropriate supports for $X$ and $Y$. Two questions:
Is $P(f,g)$ a distance function of two distributions $f$ and $g$.
Is there any relationship between this distance measure and the well-known measures?
Regarding the first question, $P(f,g)$ may not satisfy "symmetry" condition of a distance; in this case, it is enough to say distance of $f$ from $g$. This is similar to the Kullback–Leibler divergence also does not satisfy the symmetry condition, but it is a well-known distance metric for many applications.