Non-measurable sets vs. negligible sets

Question

I am interested in understanding the difference between negligible (or null) sets and non-measurable sets.

1) How do they relate to each other?

2) What does make them different?

3) Is a non-measurable set negligible?

4) Is a negligible set a non-measurable set whose measure has been set to $0$?

Answer 1

Let $(X, \cal M, \mu)$ be a measure space.

1) A set $E \subset X$ is null if there exists $F \in \cal M$ such that $E \subset F$ and $\mu(F) = 0$. A set $E \subset X$ is nonmeasurable if $E \notin \cal M$.

2) See 1). They are defined in two different ways.

3) A nonmeasurable set $E$ may or may not be null. It is null if and only if there exists a set $F$ with $E \subset F$, $F \in \cal M$, and $\mu(F) = 0$.

4) A null set could be either measurable or nonmeasurable. If $E$ is a measurable null set then necessarily $\mu(E) = 0$.

Answer 2

Non-measurable sets cannot be measured. When you say "negligible" what you really mean is that the set IS measurable, but it has measure zero. Therefore non-measurable sets are not negligible, since they cannot be measured, which makes the answer to number 4 no too.

In fact, if you have any set with positive Lebesgue measure, it can be proven that it contains a non-Lebesgue measurable set inside it. Also, if you have a Lebesgue measurable subset of a non-Lebesgue measurable set, that subset will have measure zero.

A more precise way of thinking of a "negligible" set is that it is measurable, but has measure zero. Negligible usually refers to the effect on your calculations, whether it's integration or just measuring sets, which is that there is no effect. This gives rise to the term "almost everywhere", or "almost surely" if you're talking to a probabilist, meaning a certain property is true EXCEPT on a set of measure zero.

I was under the interpretation that null and negligible were different things. When I learned measure theory, null sets were always measurable, but I suppose you could define them to be any subset of your space and still get the same results.

Answer 3

If you want to talk about general measures, see Umberto's answer. For Lebesgue measure, the relationship between non-measurable and null gets a bit nicer. So, assuming we're talking about Lebesgue measure:

All null sets and subsets of null sets are measurable. This property is called completeness of a measure. If we have a complete measure (like Lebesgue measure), the notions of null and non-measurable are completely disjoint - null sets are just measurable sets of measure zero, and non-measurable sets can't have any measure assigned to them, so, in particular, they can't be null.

To summarize and answer your questions directly:

1-2.) For Lebesgue measure, they're disjoint. Null sets are measurable sets of measure zero, and non-measurable sets are pathological sets that can't be measured and are usually ignored. Null sets are typically well-behaved in contrast.

3.) No. Any set with Lebesgue outer measure zero is measurable and null.

4.) No. We can't just take all non-measurable sets and declare them to have measure zero. If we did, then we wouldn't have a consistent way of assigning measures to sets.

We can, however, decide that any subset of a null set should still be null/measurable without causing any problems. This is called completing a measure. This is the difference between Lebesgue measure with the Borel $\sigma$-algebra, in which some subsets of null sets are non-measurable, and Lebesgue measure with the usual Lebesgue $\sigma$-algebra, which is basically just the Borel $\sigma$-algebra with some added sets to make sure that all null sets are measurable.