Equivalence of Definitions of Total Variation of Real Function on Closed Bounded Interval

From ProofWiki
Jump to navigation Jump to search

Theorem

The following definitions of the concept of Total Variation of Real Function on Closed Bounded Interval are equivalent:

Definition 1

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:

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


Definition 2

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

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

Let $\map {\PP_F} {\closedint a b}$ be the set of finite subsets of $\closedint a b$.

For each finite non-empty subset $\SS$ of $\closedint a b$, write:

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

with:

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

Also write:

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


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

$\ds \map {V_f} {\closedint a b} = \map {\sup_{\SS \mathop \in \map {\PP_F} {\closedint a b} } } {\map {V_f^\ast} {\SS ; \closedint a b} }$


Proof

We aim to prove that:

$\ds \sup_{P \mathop \in \map X {\closedint a b} } \paren {\map {V_f} {P ; \closedint a b} } = \sup_{\SS \mathop \in \map {\PP_F} {\closedint a b} } \paren {\map {V_f^\ast} {\SS ; \closedint a b} }$

We will prove that:

$\ds \sup_{P \mathop \in \map X {\closedint a b} } \paren {\map {V_f} {P ; \closedint a b} } \le \sup_{\SS \mathop \in \map {\PP_F} {\closedint a b} } \paren {\map {V_f^\ast} {\SS ; \closedint a b} }$

and:

$\ds \sup_{\SS \mathop \in \map {\PP_F} {\closedint a b} } \paren {\map {V_f^\ast} {\SS ; \closedint a b} } \le \sup_{P \mathop \in \map X {\closedint a b} } \paren {\map {V_f} {P ; \closedint a b} }$


Note that:

$\map X {\closedint a b} \subseteq \map {\PP_F} {\closedint a b}$

Note also that if $\SS$ is a finite subdivision of $\closedint a b$, we have:

$\map {V_f} {\SS ; \closedint a b} = \map {V_f^\ast} {\SS ; \closedint a b}$

So, we have:

$\set {\map {V_f} {P ; \closedint a b} : P \in \map X {\closedint a b} } \subseteq \set {\map {V_f^\ast} {\SS ; \closedint a b} : \SS \in \map {\PP_F} {\closedint a b} }$


So, from Supremum of Subset we obtain:

$\ds \sup_{P \mathop \in \map X {\closedint a b} } \paren {\map {V_f} {P ; \closedint a b} } \le \sup_{\SS \mathop \in \map {\PP_F} {\closedint a b} } \paren {\map {V_f^\ast} {\SS ; \closedint a b} }$

In Equivalence of Definitions of Bounded Variation for Real Function on Closed Bounded Interval, it is shown that for each $\SS \in \map {\PP_F} {\closedint a b}$, there exists a finite subdivision $\SS^\ast \in \map X {\closedint a b}$ such that $\SS \subseteq \SS^\ast$ and:

$\map {V_f^\ast} {\SS ; \closedint a b} \le \map {V_f} {\SS^\ast ; \closedint a b}$

which establishes:

$\ds \sup_{\SS \mathop \in \map {\PP_F} {\closedint a b} } \paren {\map {V_f^\ast} {\SS ; \closedint a b} } \le \sup_{P \mathop \in \map X {\closedint a b} } \paren {\map {V_f} {P ; \closedint a b} }$

$\blacksquare$