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A distance between two probability distributions defined by the largest possible difference they assign to the same event.
A total variation distance between probability measures and on the same measurable space is
where the supremum ranges over measurable sets .
If with densities and , then
A probability measure is a function defined on a sigma-algebra of subsets of a set such that , for every pairwise disjoint sequence , and .
A probability measure is a special case of a measure and is the key ingredient in a probability space; it assigns probabilities to events (measurable sets).
A measurable space is a pair consisting of a set and a sigma-algebra on .
Measurable spaces are the domains and codomains for measurable functions; adding a measure produces a measure space.
A measurable set in a measurable space is a subset with .
Measurable sets are precisely the subsets to which a measure assigns a value, and they determine measurable functions via preimages. The indicator function of a measurable set is a basic example of a measurable function.
For , the Bernoulli distribution is the distribution on with probability mass function
The probability mass function of a probability measure on a countable set , equipped with all its subsets, is the function