A measure-theoretic characterization of the empty set?

Question

This question relates to this question.

The question is about a (unambiguous) characterization of the empty set in measure-theoretic terms.

In effect trying to formalize the notion that if the measure of a set is zero (by any appropriate measure one can apply) it should be the empty set.

Is this correct and if so can this be formalized like this?

Answer 1

If the measure of a set $A\subset \Omega$ is zero under any sufficiently good measure $\mu$ on $\Omega$ (Lets say sufficiently good means probability measure or Radon measure), then $A=\emptyset$. However this is rather trivial, since you just need to check all dirac measures $\mu = \delta_x$ for all $x\in \Omega$.

On the other hand, if you disallow dirac measures or similiar measures, that give positive measure to a single point, you will always find zero sets that are not the empty set.

Or if you want a single measure, you need to take the counting measure $H^0$ (or some variant with weighting) which just is given by

$$H^0(A) = \mbox{ number of elements of $A$}.$$

However any such measure always $\infty$ for all uncountable sets like intervals and therefore in practice mostly useless.

Answer 2

Counter example: Let $A=\{1\}$ be a non-empty set and say $\mu$ is the Lebesgue measure, then: $$\mu(\emptyset)=0$$ But also: $$\mu(A)=0$$ So: $$\mu(\emptyset)=\mu(A)$$ even tough $A\neq\emptyset$.