Definition:Positive Part

Definition

Let $X$ be a set.

Let $f: X \to \overline \R$ be an extended real-valued function.

Then the positive part of $f$, $f^+: X \to \overline \R$, is the extended real-valued function defined by:

$\forall x \in X: \map {f^+} x := \max \set {0, \map f x}$

where the maximum is taken with respect to the extended real ordering.

That is:

$\forall x \in X: \map {f^+} x := \begin {cases} \map f x & : \map f x \ge 0 \\ 0 & : \map f x < 0 \end {cases}$

Also defined as

Some sources insist that $f$ be a real-valued function instead.

However, $\R \subseteq \overline \R$ by definition of $\overline \R$.

Thus, the definition given above incorporates this approach.

Also see

• Definition:Negative Part, the natural associate of positive part

• Results about positive parts can be found here.

Sources

• 1970: Arne Broman: Introduction to Partial Differential Equations ... (previous) ... (next): Chapter $1$: Fourier Series: $1.1$ Basic Concepts: $1.1.3$ Definitions
• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $8.7 \ \text{(v)}$