Definition:Negative Part

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Definition

Let $X$ be a set, and let $f: X \to \overline \R$ be an extended real-valued function.


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

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

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


That is:

$\forall x \in X: \map {f^-} x := \begin {cases} -\map f x & : \map f x \le 0 \\ 0 & : \map f x > 0 \end {cases}$


Hence the negative part of $f$ is actually defined as a positive function.


Also defined as

As a Real-Valued Function

Some sources insist, when defining the negative part, that $f$ be a real-valued function:

$f: X \to \R$

That is, that the codomain of $f$ includes neither the positive infinity $+\infty$ nor the negative infinity $-\infty$.


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

Thus, the main definition as provided on $\mathsf{Pr} \infty \mathsf{fWiki}$ incorporates this approach.


Hence it is still the case that:

$\forall x \in X: \map {f^-} x := \begin {cases} -\map f x & : \map f x \le 0 \\ 0 & : \map f x > 0 \end {cases}$


As a Negative Real Function

Some sources define the negative part of an extended real-valued function $f$ as:

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

That is:

$\forall x \in X: \map {f^-} x := \begin {cases} \map f x & : \map f x \le 0 \\ 0 & : \map f x > 0 \end {cases}$

Using this definition, the negative part is actually a negative function, which conforms to what feels more intuitively natural.


Also see

  • Results about negative parts can be found here.


Historical Note

When Henri Lebesgue did the initial analysis on measurable functions, he was able to establish some useful results on positive functions.

In order to extend this theory to the general real-valued function and extended real-valued function, he took advantage of the fact that it is easier to split such a function into two parts: the positive part and the negative part.

Unfortunately for intuition, while defining the negative part he changed its sign to turn it into a positive function.


Sources


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