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Math and science::Analysis::Tao, measure::02. Lebesgue measure

Lebesgue measurable sets.

This card proves the Lebesgue measurability of some useful sets.

Recall that a definition for Lebesgue measurability was introduced with the claim that it would specify a class of sets which have good properties in terms of their measure. The definition is included here as a recap.

Lebesgue measurability. Definition.

A set E⊂Rd is said to be Lebesgue measurable iff for every real ε>0 there exists exists an open set U⊂Rd containing E such that m∗(U∖E)≤ε. If E is Lebesgue measurable, we refer to m(E):=m∗(E) as the Lebesgue measure of E.

Using properties of Lebesgue outer measure, we show that the following sets meet the above criteria, and are thus Lebesgue measurable sets:

  1. Every [...] set is Lebesgue measurable.
  2. A [...] of Lebesgue measurable sets is Lebesgue measurable.
  3. Every [...] set is Lebesgue measurable.
  4. The [...] of a Lebesgue measurable set E is Lebesgue measurable.
  5. A [...] of Lebesgue measurable sets is Lebesgue measurable.

Two others are:

  • Every set of Lebesgue outer measure [...] is measurable. These sets are called [...] sets.
  • The empty set ∅ is Lebesgue measurable.

The proofs are on the reverse side.