Let $(X, \mathcal{A}, \mu)$ be a measure space, and let $s, t: X \to [0, \infty)$ be non-negative simple functions. Let $\alpha, \beta \ge 0$. Then:

1. **Linearity:** $\displaystyle\int_X (\alpha s + \beta t) \, d\mu = \alpha \int_X s \, d\mu + \beta \int_X t \, d\mu$. 2. **Monotonicity:** If $s(x) \le t(x)$ for all $x \in X$, then $\displaystyle\int_X s \, d\mu \le \int_X t \, d\mu$. 3. **Additivity over domains:** For disjoint $E, F \in \mathcal{A}$, $\displaystyle\int_{E \cup F} s \, d\mu = \int_E s \, d\mu + \int_F s \, d\mu$. 4. **Measure-valued set function:** For fixed $s$, the map $E \mapsto \int_E s \, d\mu$ defines a measure on $(X, \mathcal{A})$.