Monotonicity of measure

2.1.1 - Real Analysis: Royden

1. Let $m$ be a set function defined for all sets in a $\sigma$-algebra $\mathcal{A}$ with values in $[0,\mathrm{\infty })$. Assume $m$ is countably additive over countable disjoint collections of sets in $\mathcal{A}$. Prove that if $A$ and $B$ are two sets in $\mathcal{A}$ with $A\subseteq B$, then $m(A)\le m(B)$. This property is called monotonicity.

Proof:

Note that

and thus