Let $(X, \mathcal{A})$ be a measurable space, and let $s, t: X \to \mathbb{R}$ be simple functions. Let $\alpha \in \mathbb{R}$. Then the following functions are simple:
In particular, the collection of all simple functions on $(X, \mathcal{A})$ forms a real algebra (a vector space closed under pointwise multiplication) and a vector lattice.