What is Total Variation Distance?

Probability distributions $a$ (solid) and $b$ (shaded).
Subset $x=\{1, 3, 4\}$. $P_a(x)=5/16$ and $P_b(x)=4/16$.
$\vert\vert a-b\vert\vert$ is the largest difference across all 256 possible subsets.
$\vert\vert a-b\vert\vert = 1/2 \sum \vert a-b\vert$

[Diaconis, 1988, pages 28-30] [Rosenthal, 1995]