Monotonicity of a measure ($A\subseteq B⟹\mu (A)\le \mu (B)$)

For measurable sets $A\subseteq B$, the disjoint union $B=A\bigsqcup (B\setminus A)$ and finite additivity give $\mu (B)=\mu (A)+\mu (B\setminus A)\ge \mu (A)$. This uses nonnegativity and is valid with infinite values; it does not subtract infinities. Monotonicity of a measure implies that increasing measurable sets have nondecreasing measures, but the equality of their limit with the measure of their union uses the stronger axiom of countable additivity.