Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space. A finite sum of real numbers is that of Finite Sum Notation in a Field, formed in the field $\mathbb{R}$ over an initial segment.

Let $m$ be a natural number, let $f_{k}:X\to\mathbb{R}$ be integrable for every $k\in[m]$, and let $c_{k}$ be a real number for every $k\in[m]$. Since $1\le m$, the initial segment $[m]$ contains $1$, so the finite sums written below are defined. Let $g:X\to\mathbb{R}$ be the map given by

$$g(x)=\sum_{k=1}^{m}c_{k}f_{k}(x)\qquad(x\in X).$$

Then the following hold.

1. (Integrability)(i) $g$ is measurable and integrable.

2. (Linearity)(ii)

$$\int_{X}g\,d\mu=\sum_{k=1}^{m}c_{k}\int_{X}f_{k}\,d\mu .$$