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The positive and negative variations are nondecreasing and give the Jordan identities

Statement

For a bounded-variation function $f:[a,b]\to\mathbb R$ , the functions $P_f$ and $N_f$ are nondecreasing and

$f(x)=f(a)+P_f(x)-N_f(x),\qquad V_f(x)=P_f(x)+N_f(x).$

Both $P_f(a)$ and $N_f(a)$ are $0$ .

Facts & Assumptions

Given: A bounded-variation function $f:[a,b]\to\mathbb R$ and its functions $V_f,P_f,N_f$ .

[L1]

$V_f,P_f,N_f$ are defined by the displayed formulas in Variation function and positive and negative variations.

[L2]

For $x\le y$ , $V_f(y)-V_f(x)=\operatorname{Var}_{[x,y]}(f)$ (Total variation is additive over adjacent subintervals and decreases under restriction).

[L3]

$|f(y)-f(x)|\le\operatorname{Var}_{[x,y]}(f)$ (Total variation bounds increments; bounded-variation functions are bounded; zero variation means constant).

[L4]

A function is nondecreasing when $x\le y$ implies $g(x)\le g(y)$ (Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of $\mathbb{R}$ , with the dictionary to monotone sequences).

[L5]

$|u|\ge u$ and $|u|\ge -u$ (Absolute value in an ordered field).

Proof

technique · direct

1.1

For $x\le y$ , [L2] and [L3] give $V_f(y)-V_f(x)\ge|f(y)-f(x)|$ , hence this difference is at least both $f(y)-f(x)$ and $f(x)-f(y)$ . Therefore $P_f(y)-P_f(x)\ge0$ and $N_f(y)-N_f(x)\ge0$ , so both functions are nondecreasing.

L1L2L3L4L5

1.2

Adding and subtracting the defining formulas gives $P_f(x)+N_f(x)=V_f(x)$ and $P_f(x)-N_f(x)=f(x)-f(a)$ . At $x=a$ , $V_f(a)=0$ , so $P_f(a)=N_f(a)=0$ .

L1algebra

2.1

Rearranging the second identity in step 1.2 gives $f=f(a)+P_f-N_f$ , while the first is the asserted variation identity.

step 1.2algebra∎

Depends on

Used by

Jordan decomposition for functions of bounded variation Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 44 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Christopher Heil, Absolute Continuity and the Banach-Zaretsky Theorem (standard reference, not scraped)