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/2
n+1,a
n+e/2
n+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.