Sums, Multiples and Products of Functions into [0,∞]

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, let $X$ be a set, let $f,g:X\to[0,\infty]$ be functions and let $c\in[0,\infty]$. Sums and products in $[0,\infty]$ are those of Addition and Multiplication on [0,∞] (i) and Addition and Multiplication on [0,∞] (ii) (the three functions below exist by Maps and Relations Given by Formulas (i)).

(i) Sum. The sum of $f$ and $g$ is the function $f+g:X\to[0,\infty]$ with

$$(f+g)(x)=f(x)+g(x)\qquad(x\in X).$$

(ii) Multiple. The multiple of $f$ by $c$ is the function $cf:X\to[0,\infty]$ with

$$(cf)(x)=c\,f(x)\qquad(x\in X).$$

(iii) Product. The product of $f$ and $g$ is the function $fg:X\to[0,\infty]$ with

$$(fg)(x)=f(x)\,g(x)\qquad(x\in X).$$

(iv) Products with an indicator. For $A\subseteq X$, the indicator function $\mathbf{1}_{A}$ takes only the values $0$ and $1$, which lie in $[0,\infty]$ (The Extended Real Line, Its Order, and [0,∞] (ii)), so it is also a function into $[0,\infty]$; $f\mathbf{1}_{A}$ is the product of $f$ and $\mathbf{1}_{A}$ as in (iii):

$$(f\mathbf{1}_{A})(x)=f(x)\,\mathbf{1}_{A}(x)\qquad(x\in X).$$