We say that if is a bounded interval with endpoints and (where ), then the length of is ; this is denoted . In this sense, the length of an interval doesn’t depend on whether or not the interval is open or closed; . This idea of the length of a (bounded) interval is very intuitive, and the ideas from measure theory (specifically the Lebesgue measure on ) will tell us how to measure the “length” of more general sets than just intervals.

Recall that a covering of a set is a collection of sets of sets such that . If a subset of can be covered by intervals whose total length can be made arbitrarily small, then we say that is a null set. That is, is a null set if for any given greater than 0, there exists a sequence of intervals that covers (so ) and .

Note that all singleton sets, are null sets, since they can be covered by . This result can easily be generalized to any finite, or even countably infinite, set of elements. This result extends to countable collections of arbitrary null sets (not just singletons) as follows: If is a sequence of null sets, then their union, , is null too.

**Proof**: Suppose is a sequence of null sets. This means for each and each , there exists a sequence such that and . So, in particular, we can choose such that .

Since a countable union of countable sets is countable, we rearrange the elements of our sequences so that

, then calculate the total length of our sequence.

And so a union of countably many null sets is null.

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