I have the following definitions for a set to have measure zero and for a set to have volume, respectively:
A set $A$ has measure zero if for any $\epsilon > 0$ there is a covering $\{S_i\}_{i \in \mathbb{N}}$ such that $\sum_{i = 0}^{\infty} \mathrm{vol}(S_i) < \epsilon$.
A set $A \subseteq \mathbb{R}^n$ has volume if the characteristic function of $A$ is integrable. That is, $\int_A 1_Adx$ exists.
My questions are: 1) What is the difference between the measure of a set and the volume of a set? 2) Specifically, is having volume zero stronger than having measure zero? I have the following example: $\mathbb{Q}$ has measure zero in $\mathbb{R}^2$ (since $\mathbb{R}$ does), but $\mathbb{Q}$ does not have volume since $1_\mathbb{Q}$ is discontinuous everywhere, thus having measure zero does not imply a set has volume zero. Is there a nice example of question 2) above? 3) What is the geometric difference and meaning of volume versus measure? Is there a context in which one is not strong enough for a certain result (i.e. why bother making two definitions?) I have not had too much exposure to topology (which is where I suspect this distinction pops up) or Lebesgue integration, so simply giving an example without too must justification is okay with me.