Monotonicity of measure

Published

26 April 2024

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 $[0, \infty)$. 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 $m(A) \leq m(B)$. This property is called monotonicity.

Proof:

Note that

\[B = A \; \dot \cup \; (B\setminus A)\]

and thus

\[m(B) = m(A) + m(B\setminus A) \geq m(A)\]