1549901369-Elements_of_Real_Analysis__Denlinger_

3.4 *The Cantor Set 171

where each Cn is the union of 2n disjoint closed intervals, all of length _2_.

3n

Let € > 0. Each of the 2n closed intervals comprising Cn can be covered by

an open interval of somewhat longer length, say _2_ 3n + ~2n+ 1.

I

---I 3" + t:

311 211+1

Figure 3.18

Thus, Cn can be covered by 2n open intervals whose combined length is

Since (~)n --+ 0, we can choose n E N3 (~)n < ~· For this n, Cn can be

covered by 2n open intervals whose combined length is (~) n + ~ < c:. But

C <;:;; Cn. Therefore, C can be covered by a finite collection of open intervals of

total length less thane. Thus, Chas measure zero. •

"FAT" CANTOR-LIKE SETS OF POSITIVE MEASURE

The concept of measure zero is a special case of the more general concept
of "measurability" of sets, developed in the early twentieth century by Henri
Lebesgue and others. To present this concept here would involve complications
beyond the scope of this course. For a full treatment, the reader may consult
any real analysis book^5 containing "measure theory." It suffices to say that it
is possible to define a class M of "measurable" subsets of JR and a "measure
function" μ : M --+ JR U { +oo} such that

(μ1) VA EM, μ(A)~ 0.
(μ2) 0 and JR are measurable; μ(0) = 0 and μ(JR) = +oo.
(μ3) If A and B are measurable, then AU B, An B, and A - B are
measurable.

(μ4) The union of a countable collection {An : n E N} of measurable sets

is measurable. If the sets are pairwise disjoint thenμ CQ
1

An) = n~l μ(An)·

(If the collection is finite, then "oo" in this formula is replaced by some "n".)

5. An excellent source is Royden [116] listed in the Bibliography.