I came across the following lemma in some book:
Every countable set has measure $0$ and the proof involved "breaking" down the countable set into a countable union of points and then proved that every point has measure $0$ and hence by sub-additivity proved that a countable set has measure $0$, but am I right in assuming that this will only ever be true when we are talking about the Lebesgue measure? Because if we had instead defined the measure as a trivial measure then only the null set would have measure $0$
Best Answer
You're almost right.
This is true for any measure which gives $0$ to singletons. It could be the Lebesgue measure, or some other regular Borel measure, or a measure giving $0$ to countable sets and $\infty$ to uncountable sets.
But of course, there can be measure which give to some singleton, or some countable set a positive measure. For example the measure $\mu(A)=\begin{cases} 0 & 1\notin A\\ 1 & 1\in A\end{cases}$ is such measure.