The total variation distance between probability measures 
and 
on the same measurable space is 
, where the supremum is over measurable
sets. It equals half the total variation of the
signed measure 
on the whole space.
For probability mass functions 
and 
on a countable set, it is
 |
The distance lies between 0 and 1. It is zero precisely when the distributions agree and is 1 when they have disjoint supports. The trace distance of diagonal density matrices reduces to this classical distance.
