Positive and negative parts

In mathematics, the positive part of a real or extended real-valued function is defined by the formula $$ f^+(x) = \max(f(x),0) = \begin{cases} f(x) & \text{ if } f(x) > 0 \\ 0 & \text{ otherwise.} \end{cases} $$

Intuitively, the graph of $$f^+$$ is obtained by taking the graph of $$f$$, chopping off the part under the $f(x) = x^{2} − 4$-axis, and letting $$f^+$$ take the value zero there.

Similarly, the negative part of $x$ is defined as $$ f^-(x) = \max(-f(x),0) = -\min(f(x),0) = \begin{cases} -f(x) & \text{ if } f(x) < 0 \\ 0 & \text{ otherwise} \end{cases} $$

Note that both $f$ and $f^{+}$ are non-negative functions. A peculiarity of terminology is that the 'negative part' is neither negative nor a part (like the imaginary part of a complex number is neither imaginary nor a part).

The function $f^{&minus;}$ can be expressed in terms of $f$ and $f^{+}$ as $$ f = f^+ - f^-. $$

Also note that $$ |f| = f^+ + f^-.$$

Using these two equations one may express the positive and negative parts as $$\begin{align} f^+ &= \frac{|f| + f}{2} \\ f^- &= \frac{|f| - f}{2}. \end{align}$$

Another representation, using the Iverson bracket is $$\begin{align} f^+ &= [f>0]f \\ f^- &= -[f<0]f. \end{align}$$

One may define the positive and negative part of any function with values in a linearly ordered group.

The unit ramp function is the positive part of the identity function.

Measure-theoretic properties
Given a measurable space $f^{&minus;}$, an extended real-valued function $(X, Σ)$ is measurable if and only if its positive and negative parts are. Therefore, if such a function $f$ is measurable, so is its absolute value $f$, being the sum of two measurable functions. The converse, though, does not necessarily hold: for example, taking $|f|$ as $$f = 1_V - \frac{1}{2},$$ where $f$ is a Vitali set, it is clear that $V$ is not measurable, but its absolute value is, being a constant function.

The positive part and negative part of a function are used to define the Lebesgue integral for a real-valued function. Analogously to this decomposition of a function, one may decompose a signed measure into positive and negative parts &mdash; see the Hahn decomposition theorem.