Positive Part, Negative Part and Absolute Value of a Real-Valued Function

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, let $X$ be a set and $f:X\to\mathbb{R}$ a function. For $y\in\mathbb{R}$, $\max\{y,0\}$ is the greatest element of $\{y,0\}$ (it exists by The Maximum and Minimum of Two Elements of a Total Order (i)), and $|y|$ is the absolute value of $y$. A function into $\mathbb{R}$ with values $\ge0$ is also a function into $[0,\infty]$, which contains every real number $\ge0$ (The Extended Real Line, Its Order, and [0,∞] (ii)).

(i) Positive part. The positive part of $f$ is the function $f^{+}:X\to\mathbb{R}$ with

$$f^{+}(x)=\max\{f(x),0\}\qquad(x\in X)$$

For every $x\in X$, $f^{+}(x)\ge0$, so $f^{+}(x)\in[0,\infty]$ and $f^{+}$ is also a function into $[0,\infty]$ (The Maximum and Minimum of Two Elements of a Total Order (iii)).

(ii) Negative part. The negative part of $f$ is the function $f^{-}:X\to\mathbb{R}$ with

$$f^{-}(x)=\max\{-f(x),0\}\qquad(x\in X)$$

For every $x\in X$, $f^{-}(x)\ge0$, so $f^{-}(x)\in[0,\infty]$ and $f^{-}$ is also a function into $[0,\infty]$ (The Maximum and Minimum of Two Elements of a Total Order (iii)).

(iii) Absolute value. The absolute value of $f$ is the function $|f|:X\to\mathbb{R}$ with

$$|f|(x)=|f(x)|\qquad(x\in X)$$

($|f|$ is also a function into $[0,\infty]$, by Arithmetic and Order Rules in the Real Numbers, Stated for ℝ (ix)).