prove that any singleton,finite or countable set is measurable.

Question

I have searched a lot but I am unable to get its proof from any where.Can any one help me with its proof.I know what a measurable set is and also many of its application but at this I am just stuck

Answer 1

This statement is false for a general measure. Consider lebesgue measure restricted to the $\sigma$ algebra $\{\phi, \mathbb{R}\}$. Then singletons are not measurable.

However, say you are trying to prove this for lebesgue measure with its full $\sigma$ algebra. You could go about the proof in the following way:

1) Use properties of the sigma algebra to show singletons are measurable.

2) Use properties of the sigma algebra to show this implies countable sets are measurable.