Partial Sums of a Sequence of Functions into [0,∞]

In the setting of Measurable Functions and the Lebesgue Integral: Notation, let $X$ be a set, let the sum $+$ on $[0,\infty]$ be as in Addition and Multiplication on [0,∞] (i), and let $(f_{k})_{k\in\mathbb{N}}$ be a sequence of functions $X\to[0,\infty]$, that is, a sequence in the set $[0,\infty]^{X}$ of functions from $X$ to $[0,\infty]$ (The Elements of the Class of Functions from a Set to a Set, and This Class Is a Set (ii), The Elements of the Class of Functions from a Set to a Set, and This Class Is a Set (i)). For $x\in X$, $(f_{k}(x))_{k}$ is the sequence in $[0,\infty]$ of Evaluating a Sequence of Maps at a Point Gives a Sequence, and Its Range (i).

(i) Partial-sum functions. There is a sequence $(H_{n})_{n\in\mathbb{N}}$ of functions $X\to[0,\infty]$ with

$$H_{n}(x)=\sum_{k=1}^{n}f_{k}(x)\qquad(n\in\mathbb{N},\ x\in X),$$

where the right-hand side is the $n$-th partial sum in $[0,\infty]$ of the sequence $(f_{k}(x))_{k}$: for all $n\in\mathbb{N}$ and $x\in X$, and every sequence $u$ in $[0,\infty]$ with $u_{k}=f_{k}(x)$ for every $k\in\mathbb{N}$, $H_{n}(x)$ is the $n$-th partial sum $\sum_{k=1}^{n}u_{k}$ of $u$.