Definition:Positive Part
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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): $8.7 \ \text{(v)}$