Union with a zero measure set does not alter the measure

Solution to Measure, Integration & Real Analysis by Axler Section 2A Exercise 2A1

Prove that if $A$ and $B$ are subsets of $\mathbf R$ and $|B|=0$, then $|A\cup B|=|A|$.

Solution: Because $A\subset A\cup B$, by 2.5 we have \begin{equation}\label{2a1.1}|A|\leqslant |A\cup B|.\end{equation}By 2.8 and $|B|=0$, we also have \begin{equation}\label{2a1.2}|A\cup B|\leqslant |A|+|B|=|A|.\end{equation}Combining \eqref{2a1.1} and \eqref{2a1.2}, we conclude that $|A\cup B|=|A|$.