1. ## countable set

2. Originally Posted by Kat-M
With respect to a particular measure?

For the Lebesgue measure the answer is no, since any countable set is measurable with measure 0.

3. Originally Posted by pomp
With respect to a particular measure?

For the Lebesgue measure the answer is no, since any countable set is measurable with measure 0.
with respect to any measure. so for a different measure, there is a countable set which is not measurable?

4. Originally Posted by Kat-M
with respect to any measure. so for a different measure, there is a countable set which is not measurable?
My intuition says no, it's my understanding that any countable set has measure zero meaning it must be measurable, but the only measure I have ever worked with is the Borel and Lebesgue measures. Measure Theory is not really my thing, and it's also 7:48 and I haven't slept yet I could be wrong.

I'm sorry, but I'm sure your question will be answered very swiftly by someone more confident than me!

5. Hello,

There are many things you have to define before talking about measurability !

A counterexample is :

Let's deal with $(\mathbb{R},\mathcal{A},\lambda)$, where $\mathcal{A}=\{\mathbb{R},\emptyset,\{0\},\mathbb{R }\backslash \{0\}\}$ and $\lambda$ is the Lebesgue measure.
The only measurable sets are the ones in $\mathcal{A}$.
Hence $\mathbb{N}$, which is a countable set, is not measurable !