Positive and negative parts

In mathematics, the positive part of a real or extended real-valued function is defined by the formula

f^%2B(x) = \max(f(x),0) = \begin{cases} f(x) & \mbox{ if } f(x) > 0 \\ 0 & \mbox{ otherwise.} \end{cases}

Intuitively, the graph of f^%2B is obtained by taking the graph of f, chopping off the part under the x-axis, and letting f^%2B take the value zero there.

Similarly, the negative part of f is defined as

f^-(x) = -\min(f(x),0) = \begin{cases} -f(x) & \mbox{ if } f(x) < 0 \\ 0 & \mbox{ 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 can be expressed in terms of f+ and f− as

f = f^%2B - f^-. \,

Also note that

|f| = f^%2B %2B f^-\,.

Using these two equations one may express the positive and negative parts as

f^%2B= \frac{|f| %2B f}{2}\,
f^-= \frac{|f| - f}{2}.\,

Another representation, using the Iverson bracket is

f^%2B= [f>0]f\,
f^-= -[f<0]f.\,

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

Measure-theoretic properties

Given a measurable space (X,Σ), an extended real-valued function f 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-{1\over2},

where V is a Vitali set, it is clear that f 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 — see the Hahn decomposition theorem.

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