Total variation

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In mathematics, the total variation of a real-valued function f on the bounded interval [a, b] is

<math>\sup_P \sum_i | f(x_{i+1})-f(x_i) | </math>

the supremum running over all partitions P = { x₁, ..., x_n } of the interval [a, b]. In effect, the total variation is the vertical component of the arc-length of the graph of f. The function f is said to be of bounded variation precisely if the total variation of f is finite.

Total variation distance in probability theory

In probability theory, the total variation distance between two probability measures P and Q on a sigma-algebra F is

<math>\sup\left\{\,\left|P(A)-Q(A)\right| : A\in F\,\right\}.</math>

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

Total variation in measure theory

Given a signed measure μ on a measurable space (X,Σ), and its Hahn-Jordan decomposition into the difference of two non-negative measures

<math>\mu=\mu^+-\mu^-,</math>

its variation is the non-negative measure

<math>|\mu|=\mu^++\mu^-,</math>

and its total variation is defined as

<math>\|\mu\|=|\mu|(X).</math>de:Variation (Mathematik)