Total variation distance of probability measures

In probability theory, the total variation distance is a distance measure for probability distributions. It is an example of a statistical distance metric, and is sometimes just called "the" statistical distance.

The total variation distance between two probability measures P and Q on a sigma-algebra ${\displaystyle {\mathcal {F}}}$ of subsets of the sample space ${\displaystyle \Omega }$ is defined via

Informally, this is the largest possible difference between the probabilities that the two probability distributions can assign to the same event.

For a finite or countable alphabet we can relate the total variation distance to the 1-norm of the difference of the two probability distributions as follows:

Similarly, for arbitrary sample space ${\displaystyle \Omega }$, measure ${\displaystyle \mu }$, and probability measures ${\displaystyle P}$ and ${\displaystyle Q}$ with Radon-Nikodym derivatives ${\displaystyle f_{P}}$ and ${\displaystyle f_{Q}}$ with respect to ${\displaystyle \mu }$, an equivalent definition of the total variation distance is

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