Lebesgue Measure

OUTER-MEASURE

1. For a set A of real numbers, consider all countable collections of open, bounded intervals that cover A, in the sense that For each such collection, consider the sum of the lengths of the intervals in the collection. The outer-measure of , , is defined to be the infimum of all such sums, that is,
2. outer-measure is monotone, and:
3. Lemma 1 (von Neumann) For a bounded set , define its integral count, , to be the number of integers in . For each , define the Then for each bounded interval ,
4. Proposition 2 If the bounded interval I is covered by a finite collection of bounded intervals, then
5. Proposition 3 The outer-measure of an interval is its length
6. Proposition 4 Outer-measure is translation invariant, in the sense that for any set and any , if ,
7. Proposition 5 Outer-measure is countably monotone, in the sense that if is any countable collection of sets, disjoint or not, that covers a set , then

THE σ-ALGEBRA OF LEBESGUE MEASURABLE SETS

1. Definition : set is said to be Lebesgue measurable, or simply measurable, provided that for any set , The collection of measurable sets is denoted by
2. Proposition 6 If , is measurable and , then
3. Proposition 7 Any set of outer-measure zero is measurable
4. Proposition 8 The translate of a measurable set is measurable
5. Proposition 9 The union of a finite collection of measurable sets is measurable
6. Proposition 10 If is any set and is a finite, disjoint collection of measurable sets, then In particular,
7. Definition countable, disjoint collection of measurable subsets of is called a measurable partition of provided that .
8. Lemma 11 Let, a countable union of measurable sets. Then there is a measurable partition of for which each .
9. Proposition 12 The union of a countable collection of measurable sets is measurable
10. Proposition 13 Every interval is measurable
11. A collection of subsets of is called an provided that it is an algebra and is closed with respect to countable unions and countable intersections.
12. A set of real numbers is said to be a set provided that it is the intersection of a countable collection of open sets and said to be an set provided that it is the union of a countable collection of closed sets
13. Since M is a , every set and every set is measurable
14. The B is defined to be the smallest that contains all open sets. Since is such a
15. Theorem 14 The collection of measurable sets is a that contains the and all sets of outer-measure zero.

FINER PROPERTIES OF MEASURABLE SETS

1. Definition The restriction of the set-function outer-measure to the of measurable sets is called Lebesgue measure or simply measure. It is denoted by , so that if is a measurable set, its Lebesgue measure, , is defined by
2. Theorem 15 (the Regularity of Lebesgue Measure) If is Lebesgue measurable and , then there is a closed set and an open set for which
3. Corollary 16 If is measurable, then there is a set and an set for which
4. Corollary 17 A set of real numbers is measurable if and only if it is a set from which a set of measure zero has been excised
5. Theorem 18 The collection of Lebesgue measurable, sets is the smallest that contains the and all sets of outer-measure zero.
6. Theorem 19 If and then there is a finite, disjoint collection of open, bounded intervals for which, if , then

COUNTABLE ADDITIVITY AND CONTINUITY OF MEASURE, AND THE BOREL-CANTELLI LEMMA

1. Theorem 20 (the Countable Additivity of Measure) Lebesgue measure is countably additive, in the sense that if is a measurable partition of , then
2. Theorem 21 The set-function Lebesgue measure, defined on the of Lebesgue measurable sets , assigns length to any interval, is translation invariant, and is countably additive A countable collection of sets is said to be ascending provided that for each , , and said to be descending provided that for each ,
3. Theorem 22 (the Continuity of Measure)
   • If is an ascending collection of measurable sets, then
   • If is a descending collection of measurable sets and , then
4. The Borel-Cantelli Lemma:If is a countable collection of measurable sets for which, then almost all belong to at most finitely many of the

VITALI’S EXAMPLE OF A NON-MEASURABLE SET

1. Theorem 23 (Vitali) If is a set of real numbers for which , then there is a subset of that is not measurable.
2. Corollary 24 There are disjoint sets of real numbers and for which

THE CANTOR SET AND THE CANTOR-LEBESGUE FUNCTION