Define metric between two finite probability distribution functions Suppose I have two finite probability distributions $X$ and $Y$. What I am trying to find is a good metric that will tell me how close they are too each other. That is, some $m(X,Y)\in[0,1]$ such that $0$ means the distributions are the same while $1$ means they are "completely different" (the meaning of "completely different" can be open to multiple interpretations). For example, suppose
$$X=\{.1,.3,.1,.5\}$$
$$Y=\{.15,.25,.05,.55\}$$
Then the metric should output something small since these are nearly the same distribution. One example I came up with was
$$m(X,Y)=\sum_{i=1}^n\frac{|x_i-y_i|^p}{2}$$
for any $p\geq 1$. For the example above and $p=1$, we would get
$$m(X,Y)=\frac{1}{2}(|.1-.15|+|.3-.25|+|.1-.05|+|.5-.55|)$$
$$=\frac{1}{2}(.05+.05+.05+.05)=.1$$
This works, but it seems rather clunky and non-standard. I know about the Kolmogorov-Smirnov test, but this is used to test whether the distributions are the same, and not define a metric measuring how close they are.
Does anybody know a standard way to define such a metric? If not, are there any phrases/topics I could investigate to find such a metric?
 A: There is the notion of the total variation distance between probability measures $\mathbb P$ and $\mathbb Q$ on a $\sigma$-algebra $\mathcal F$, defined by
$$
\delta(\mathbb P,\mathbb Q) = \sup_{A\in\mathcal F}|\mathbb P(A) - \mathbb Q(A)|.
$$
When the underlying sample space $\Omega$ is countable, then we have the equivalence
$$
\delta(\mathbb P,\mathbb Q) = \frac12\sum_{\omega\in\Omega}|\mathbb P(\omega)-\mathbb Q(\omega)|.
$$
So in the example where
$$
 P(X=0) = \frac1{10},\quad  P(X=1) = \frac3{10},\quad  P(X=2)=\frac1{10},\quad  P(X=3)=\frac12
$$
and
$$
 P(Y=0) = \frac3{20},\quad  P(Y=1) = \frac14,\quad  P(Y=2)=\frac1{20},\quad  P(Y=3)=\frac{11}{20}
$$
we can think of this as $\Omega=\{0,1,2,3\}$ where $\mathbb P(\omega) = P(X=\omega)$ and $\mathbb Q(\omega)=P(Y=\omega)$. So we have
\begin{align}
\delta(\mathbb P,\mathbb Q) &= \frac12\sum_{\omega\in\Omega}|\mathbb P(\omega)-\mathbb Q(\omega)|\\
&= \frac12\left(\left|\frac1{10}-\frac3{20}\right| + \left|\frac3{10}-\frac14\right| + \left|\frac1{10}-\frac1{20}\right| + \left|\frac12-\frac{11}{20}\right| \right)\\
&= \frac1{10}.
\end{align}
