What's the difference between a null set and an empty set? From my notes, I understand that an empty set always has a null size (is this different from saying that it has a size = 0?).
What are then examples of non-empty null sets?
Thanks for your time.
--- Edit: There seems to be some conceptual difference. Wikipedia also has 2 different articles (which I mostly don't understand :-) ) (https://en.wikipedia.org/wiki/Null_set and https://en.wikipedia.org/wiki/Empty_set).
 A: In analysis and measure theory, the term null set is also used to denote a set which has "size" zero, but in that case, size means a different thing. For instance, on the real line, it is customary to use length (at least in naïve settings). So the interval $[-3, 5]$ has size $8$ and the set $(2, 3)\cup [5, 7]$ has size $3$.
In this setting, examples of non-empty null-sets are:


*

*Single-element sets, like $\{5\}$.

*In fact, any countable set, like $\Bbb Q$

*The Cantor set
So whether there is a difference between the phrases "empty set" and "null set" depends entirely on the context.
A: This depends on the context.
In the context of set theory, the null set is the empty set. And that's the end of it.
In the context of measure theory, analysis, or probability, a null set is a set whose measure is $0$. For example in the usual Borel measure, finite sets are null sets; countable sets are null sets; and even some uncountable sets (e.g. the Cantor set) are null sets. But they are certainly not empty.
In that context, a null set is a set which is completely uninteresting "for practical purposes" and we can ignore safely ignore it if we choose to. So this statement is more general than just "empty".
Note, however, that if you define an equivalence relation "$A\sim B$ if and only if $A\mathbin{\triangle}B$ is a null set", then the null sets are exactly those equivalent to the empty set.
