Null set

A null subset N of R (not to be confused with the empty set) is one satisfying the following condition:
Given any strictly positive number e, there is a sequence {I_n} of open intervals (a_n,b_n) such that N is contained in the union of the I_n and the total length of the I_n is less than e.
Null sets are difficult to grasp intuitively (at least to the present author). As the name suggests they are in a sense "negligible" (see below for an illustration of this point), but we can also prove
THEOREM: Any countable subset of R is null
The idea of the proof is to list the set as, say, {a_n} and then (given e>0) enclose a_n in the interval (a_n-e/2ⁿ⁺¹,a_n+e/2ⁿ⁺¹)
Thus, although there are infinitely many rational numbers between any two distinct real numbers, it is possible to cover the rational numbers in a sequence of intervals of total length less than 0.00001.
Null sets play a key role in the definition of the Lebesgue integral: if functions f and g agree almost everywhere then f is integrable iff g is, and then their integrals are equal.