Let $X$ and $Y$ be two continuous random variables with common support and with PDF $f(x)$ and $g(y)$. For any constant $b$ with the support of $X$ and $Y$, and  any constant $0 \leq \alpha \leq 1$, we define
\begin{equation}
D_{\alpha,\beta}(f \| g) = 1 - \alpha \Pr[X \leq b \mid Y>\beta ] - ( 1- \alpha) \Pr[X > b \mid Y\leq\beta ] \:, 
\tag{1}\end{equation}
as a measure of distance of $f$ from $g$ at point $\beta$. Two questions: 

1) Is $D_{\alpha,\beta}(f \| g)$ a divergence of distribution $f$ from $g$ at point $\beta$?  

2) Is there any relationship between (1) and the well-known distance (or divergence) measures? 

*Definition of statistical divergence: https://en.wikipedia.org/wiki/Divergence_(statistics) 

*Definition of statistical distance: https://en.wikipedia.org/wiki/Statistical_distance