Measure, Integration & Real Analysis, 2021a

Section 2D Lebesgue Measure 55
We previously defined Lebesgue measure as outer measure restricted to the Borel
sets (see 2.69). The term Lebesgue measure is sometimes used in mathematical
literature with the meaning as we previously defined it and is sometimes used with
the following meaning.
2.73 Definition Lebesgue measure
Lebesgue measure is the measure on (R, L), where L is the σ-algebra of Lebesgue
measurable subsets of R, that assigns to each Lebesgue measurable set its outer
measure.
The two definitions of Lebesgue measure disagree only on the domain of the
measure—is the σ-algebra the Borel sets or the Lebesgue measurable sets? You
may be able to tell which is intended from the context. In this book, the domain is
specified unless it is irrelevant.
If you are reading a mathematics paper and the domain for Lebesgue measure
is not specified, then it probably does not matter whether you use the Borel sets
or the Lebesgue measurable sets (because every Lebesgue measurable set differs
from a Borel set by a set with outer measure 0, and when dealing with measures,
what happens on a set with measure 0 usually does not matter). Because all sets that
arise from the usual operations of analysis are Borel sets, you may want to assume
that Lebesgue measure means outer measure on the Borel sets, unless what you are
reading explicitly states otherwise.
A mathematics paper may also refer to
a measurable subset of R, without further
explanation. Unless some other σ-algebra
is clear from the context, the author probably
means the Borel sets or the Lebesgue
measurable sets. Again, the choice probably
does not matter, but using the Borel
sets can be cleaner and simpler.
The emphasis in some textbooks on
Lebesgue measurable sets instead of
Borel sets probably stems from the
historical development of the subject,
rather than from any common use of
Lebesgue measurable sets that are
not Borel sets.
Lebesgue measure on the Lebesgue measurable sets does have one small advantage
over Lebesgue measure on the Borel sets: every subset of a set with (outer) measure
0 is Lebesgue measurable but is not necessarily a Borel set. However, any natural
process that produces a subset of R will produce a Borel set. Thus this small
advantage does not often come up in practice.
Cantor Set and Cantor Function
Every countable set has outer measure 0 (see 2.4). A reasonable question arises
about whether the converse holds. In other words, is every set with outer measure
0 countable? The Cantor set, which is introduced in this subsection, provides the
answer to this question.
The Cantor set also gives counterexamples to other reasonable conjectures. For
example, Exercise 17 in this section shows that the sum of two sets with Lebesgue
measure 0 can have positive Lebesgue measure.
Measure, Integration & Real Analysis, by Sheldon Axler