Definition:Total Variation

Real Function

Closed Bounded Interval

Let $a, b$ be real numbers with $a < b$.

Let $f: \closedint a b \to \R$ be a function of bounded variation.

Let $\map X {\closedint a b}$ be the set of finite subdivisions of $\closedint a b$.

For each $P \in \map X {\closedint a b}$, write:

$P = \set {x_0, x_1, \ldots, x_n}$

with:

$a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n = b$

Also write:

$\ds \map {V_f} {P ; \closedint a b} = \sum_{i \mathop = 1}^n \size {\map f {x_i} - \map f {x_{i - 1} } }$

We define the total variation $\map {V_f} {\closedint a b}$ of $f$ on $\closedint a b$ by:

$\ds \map {V_f} {\closedint a b} = \map {\sup_{P \mathop \in \map X {\closedint a b} } } {\map {V_f} {P ; \closedint a b} }$

Closed Unbounded Interval

Let $I$ be an unbounded closed interval or $\R$.

Let $f: I \to \R$ be a real function.

Let $\map {\PP_F} I$ be the set of finite subsets of $I$.

For each finite non-empty subset $\SS$ of $I$, write:

$\SS = \set {x_0, x_1, \ldots, x_n}$

with:

$x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n$

Also write:

$\ds \map {V_f^\ast} {\SS; I} = \sum_{i \mathop = 1}^n \size {\map f {x_i} - \map f {x_{i - 1} } }$

We define the total variation $\map {V_f} I$ of $f$ on $I$ by:

$\ds \map {V_f} I = \sup_{\SS \mathop \in \map {\PP_F} I} \paren {\map {V_f^\ast} {\SS; I} }$

Measure Theory

Signed Measure

Let $\struct {X, \Sigma}$ be a measurable space.

Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.

Let $\size \mu$ be the variation of $\mu$.

We define the total variation $\norm \mu$ of $\mu$ by:

$\norm \mu = \map {\size \mu} X$

Complex Measure

Let $\struct {X, \Sigma}$ be a measurable space.

Let $\mu$ be a complex measure on $\struct {X, \Sigma}$.

Let $\cmod \mu$ be the variation of $\mu$.

We define the total variation $\norm \mu$ of $\mu$ by:

$\norm \mu = \map {\cmod \mu} X$