The (pre-)measure of a set is no more than the sum of the (pre-)measures of the elements of a covering for that set/Statement

Statement

Suppose that $\mu$ is either a measure (or a pre-measure) on the $\sigma$-ring (or ring), $\mathcal{R}$ then:

• for all $A\in\mathcal{R}$ and for all countably infinite or finite sequences $(A_i)\subseteq\mathcal{R}$ we have:
  • $A\subseteq\bigcup_i A_i\implies\mu(A)\le\sum_{i}\mu(A_i)$

Note: this is slightly different to sigma-subadditivity (or subadditivity) which states that $\mu\left(\bigcup_i A_i\right)\le\sum_i\mu(A_i)$ (for a pre-measure, we would require $\bigcup_i A_i\in\mathcal{R}$ which isn't guaranteed for countably infinite sequences)