Definition:Total Variation/Real Function/Closed Unbounded Interval

Definition

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:

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

Sources

• 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $4.4$: Functions of Finite Variation