Real Analysis
September 14, 2003
The textbook for the course is: Folland, G. Real analysis. Modern techniques and their applications. Second Edition. Some alternative textbooks are: • Wheeden and Zygmund. An introduction to real analysis. Pure and Ap- plied Mathematics, Vol. 43. • Browder. Mathematical analysis. An introduction. Undergraduate Texts in Mathematics. • Bruckner, Bruckner and Thompson. • Federer. Geometeric measure theory.
1 Measure Theory
We wish to extend the notion of area, length or volume to the broadest possible class of sets. There is no problem in doing this with polygons and smooth curves. The difficulty comes when we deal with “pathological sets”, e.g., the Cantor set.
1.1 Pathological Examples The Cantor set is is defined as follows. Beginning with the closed unit interval, 1 2 [0, 1] delete the middle third, leaving [0, 1]\ 3 , 3 . This divides the interval into 1 2 two intervals: 0, 3 ∪ 3 , 1 . Continue this process on each of these intervals, leaving: 1 2 1 2 7 8 0, ∪ , ∪ , ∪ , 1 . 9 9 3 3 9 9 Continue this process recursively on each of these intervals. What remains is the Cantor set. What should the measure of this set be? It is zero. We will see this later. We can also form a two-dimensional analog of the Cantor set by dividing the square [0, 1] × [0, 1] into nine squares and deleting the one in the center, then proceeding recursively on the remaining blocks as in the one-dimensional case.
1 1.2 Definition of Measure We want a measure to have the following property: the measure of the union of disjoint sets should be the sum of the measures of the individual sets. Definition 1. If X is a set, define P(X) = 2X to be the collection of all subsets of X. Let A be a subset of P(X). A is called an algebra of subsets of X if 1. for all A, B ∈ A we have A ∪ B ∈ A and A ∩ B ∈ A and 2. if A ∈ A then the complement, Ac = X \ A is also in A. Remarks: 1. If A is an algebra (containing at least one set) then ∅ and X are both in A because ∅ = A ∩ Ac and X = ∅c.
2. If A1,...,An ∈ A then n n \ [ Aj and Aj j=1 j=1 are both in A as well. Definition 2. A set A ⊂ 2X is called a σ-algebra if A is an algebra and for any countable collection A1,A2,...,An,... ∈ A we have
∞ [ Ak ∈ A. k=1
If A is a σ-algebra and A1,A2,... ∈ A then the countable intersection
∞ ∞ !c \ [ c Ak = Ak k=1 k=1 is also in A. Definition 3. Let X be a set, A a σ-algebra on X and µ : A → [0, ∞] such that
∞ ! ∞ [ X µ Ak = µ (Ak) k=1 k=1 for any countable disjoint collection of sets Ak ∈ A. The function µ is said to have countable additivity. (The collection is called disjoint if Aj ∩ Ak is empty for all j 6= k.) Countable additivity implies finite additivity. That is, for a finite collection of disjoint sets, A1,...,An ∈ A, we have
n ! n [ X µ Ak = µ (Ak) . k=1 k=1
2 1.3 Examples of Measures 1. δ-measure. Let X be a set containing a point x. Take A to be the powerset 2X and define ( 0 x 6∈ A δx(A) = 1 x ∈ A.
2. Counting measure. Let X = N be the set of natural numbers and let A be 2X . Define µ(A) to be the number of elements of A.
What about length in the real line? We know how to compute the lengths of intervals, but intervals do not form a σ-algebra. Can we find a measure on R that coincides with length on intervals?
3