Measure Theory V. Liskevich 1998 1 Introduction We always denote by X our universe, i.e. all the sets we shall consider are subsets of X. Recall some standard notation. 2X everywhere denotes the set of all subsets of a given set X. If A ∩ B = ∅ then we often write A t B rather than A ∪ B, to underline the disjointness. The complement (in X) of a set A is denoted by Ac. By A4B the symmetric difference of A and B is denoted, i.e. A 4 B = (A \ B) ∪ (B \ A). Letters i, j, k always denote positive integers. The sign is used for restriction of a function (operator etc.) to a subset (subspace). 1.1 The Riemann integral Recall how to construct the Riemannian integral. Let f :[a, b] → R. Consider a partition π of [a, b]: a = x0 < x1 < x2 < . < xn−1 < xn = b and set ∆xk = xk+1 − xk, |π| = max{∆xk : k = 0, 1, . , n − 1}, mk = inf{f(x): x ∈ [xk, xk+1]},Mk = sup{f(x): x ∈ [xk, xk+1]}. Define the upper and lower Riemann— Darboux sums n−1 n−1 X X s(f, π) = mk∆xk, s¯(f, π) = Mk∆xk. k=0 k=0 One can show (the Darboux theorem) that the following limits exist Z b lim s(f, π) = sup s(f, π) = fdx |π|→0 π a Z b lim s¯(f, π) = inf s¯(f, π) = fdx. |π|→0 π a 1 Clearly, Z b Z b s(f, π) ≤ fdx ≤ fdx ≤ s¯(f, π) a a for any partition π. The function f is said to be Riemann integrable on [a, b] if the upper and lower integrals are equal. The common value is called Riemann integral of f on [a, b]. The functions cannot have a large set of points of discontinuity. More presicely this will be stated further. 1.2 The Lebesgue integral It allows to integrate functions from a much more general class. First, consider a very useful example. For f, g ∈ C[a, b], two continuous functions on the segment [a, b] = {x ∈ R : a 6 x 6 b} put ρ1(f, g) = max |f(x) − g(x)|, a6x6b Z b ρ2(f, g) = |f(x) − g(x)|dx. a Then (C[a, b], ρ1) is a complete metric space, when (C[a, b], ρ2) is not. To prove the latter ∞ statement, consider a family of functions {ϕn}n=1 as drawn on Fig.1. This is a Cauchy sequence with respect to ρ2. However, the limit does not belong to C[a, b]. 2 6 L L L L L L L L L - 1 1 1 1 1 1 − 2 − 2 + n 2 − n 2 Figure 1: The function ϕn. 2 Systems of Sets Definition 2.1 A ring of sets is a non-empty subset in 2X which is closed with respect to the operations ∪ and \. Proposition. Let K be a ring of sets. Then ∅ ∈ K. Proof. Since K 6= ∅, there exists A ∈ K. Since K contains the difference of every two its elements, one has A \ A = ∅ ∈ K. Examples. 1. The two extreme cases are K = {∅} and K = 2X . 2. Let X = R and denote by K all finite unions of semi-segments [a, b). Definition 2.2 A semi-ring is a collection of sets P ⊂ 2X with the following properties: 1. If A, B ∈ P then A ∩ B ∈ P; 3 2. For every A, B ∈ P there exists a finite disjoint collection (Cj) j = 1, 2, . , n of sets (i.e. Ci ∩ Cj = ∅ if i 6= j) such that n G A \ B = Cj. j=1 Example. Let X = R, then the set of all semi-segments, [a, b), forms a semi-ring. Definition 2.3 An algebra (of sets) is a ring of sets containing X ∈ 2X . Examples. 1. {∅,X} and 2X are the two extreme cases (note that they are different from the corresponding cases for rings of sets). 2. Let X = [a, b) be a fixed interval on R. Then the system of finite unions of subin- tervals [α, β) ⊂ [a, b) forms an algebra. 3. The system of all bounded subsets of the real axis is a ring (not an algebra). Remark. A is algebra if (i) A, B ∈ A =⇒ A ∪ B ∈ A, (ii) A ∈ A =⇒ Ac ∈ A. Indeed, 1) A ∩ B = (Ac ∪ Bc)c; 2) A \ B = A ∩ Bc. Definition 2.4 A σ-ring (a σ-algebra) is a ring (an algebra) of sets which is closed with respect to all countable unions. Definition 2.5 A ring (an algebra, a σ-algebra) of sets, K(U) generated by a collection of sets U ⊂ 2X is the minimal ring (algebra, σ-algebra) of sets containing U. In other words, it is the intersection of all rings (algebras, σ-algebras) of sets containing U. 4 3 Measures Let X be a set, A an algebra on X. Definition 3.1 A function µ: A −→ R+ ∪ {∞} is called a measure if 1. µ(A) > 0 for any A ∈ A and µ(∅) = 0; 2. if (Ai)i 1 is a disjoint family of sets in A ( Ai ∩ Aj = ∅ for any i 6= j) such that F∞ > i=1 Ai ∈ A, then ∞ ∞ G X µ( Ai) = µ(Ai). i=1 i=1 The latter important property, is called countable additivity or σ-additivity of the measure µ. Let us state now some elementary properties of a measure. Below till the end of this section A is an algebra of sets and µ is a measure on it. 1. (Monotonicity of µ) If A, B ∈ A and B ⊂ A then µ(B) 6 µ(A). Proof. A = (A \ B) t B implies that µ(A) = µ(A \ B) + µ(B). Since µ(A \ B) ≥ 0 it follows that µ(A) ≥ µ(B). 2. (Subtractivity of µ). If A, B ∈ A and B ⊂ A and µ(B) < ∞ then µ(A \ B) = µ(A) − µ(B). Proof. In 1) we proved that µ(A) = µ(A \ B) + µ(B). If µ(B) < ∞ then µ(A) − µ(B) = µ(A \ B). 3. If A, B ∈ A and µ(A ∩ B) < ∞ then µ(A ∪ B) = µ(A) + µ(B) − µ(A ∩ B). Proof. A ∩ B ⊂ A, A ∩ B ⊂ B, therefore A ∪ B = (A \ (A ∩ B)) t B. Since µ(A ∩ B) < ∞, one has µ(A ∪ B) = (µ(A) − µ(A ∩ B)) + µ(B). 5 S∞ 4. (Semi-additivity of µ). If (Ai)i≥1 ⊂ A such that i=1 Ai ∈ A then ∞ ∞ [ X µ( Ai) 6 µ(Ai). i=1 i=1 Proof. First let us proove that n n [ X µ( Ai) 6 µ(Ai). i=1 i=1 Note that the family of sets B1 = A1 B2 = A2 \ A1 B3 = A3 \ (A1 ∪ A2) ... n−1 [ Bn = An \ Ai i=1 Fn Sn is disjoint and i=1 Bi = i=1 Ai. Moreover, since Bi ⊂ Ai, we see that µ(Bi) ≤ µ(Ai). Then n n n n [ G X X µ( Ai) = µ( Bi) = µ(Bi) ≤ µ(Ai). i=1 i=1 i=1 i=1 Now we can repeat the argument for the infinite family using σ-additivity of the measure. 3.1 Continuity of a measure Theorem 3.1 Let A be an algebra, (Ai)i≥1 ⊂ A a monotonically increasing sequence of S sets (Ai ⊂ Ai+1) such that i≥1 ∈ A. Then ∞ [ µ( Ai) = lim µ(An). n→∞ i=1 S∞ Proof. 1). If for some n0 µ(An0 ) = +∞ then µ(An) = +∞ ∀n ≥ n0 and µ( i=1 Ai) = +∞. 2). Let now µ(Ai) < ∞ ∀i ≥ 1. 6 Then ∞ [ µ( Ai) = µ(A1 t (A2 \ A1) t ... t (An \ An−1) t ...) i=1 ∞ X = µ(A1) + µ(Ak \ Ak−1) k=2 n X = µ(A1) + lim (µ(Ak) − µ(Ak−1)) = lim µ(An). n→∞ n→∞ k=2 3.2 Outer measure Let a be an algebra of subsets of X and µ a measure on it. Our purpose now is to extend µ to as many elements of 2X as possible. An arbitrary set A ⊂ X can be always covered by sets from A, i.e. one can always find S∞ E1,E2,... ∈ A such that i=1 Ei ⊃ A. For instance, E1 = X, E2 = E3 = ... = ∅. Definition 3.2 For A ⊂ X its outer measure is defined by ∞ ∗ X µ (A) = inf µ(Ei) i=1 where the infimum is taken over all A-coverings of the set A, i.e. all collections (Ei),Ei ∈ S A with i Ei ⊃ A. Remark. The outer measure always exists since µ(A) > 0 for every A ∈ A. Example. Let X = R2, A = K(P), -σ-algebra generated by P, P = {[a, b) × R1}. Thus A consists of countable unions of strips like one drawn on the picture. Put µ([a, b)× 1 2 2 R ) = b − a. Then, clearly, the outer measure of the unit disc x + y 6 1 is equal to 2. The same value is for the square |x| 6 1, |y| 6 1. Theorem 3.2 For A ∈ A one has µ∗(A) = µ(A). In other words, µ∗ is an extension of µ. ∗ Proof. 1.A is its own covering. This implies µ (A) 6 µ(A). 2. By definition of infimum, for any ε > 0 there exists a A-covering (Ei) of A such that P ∗ i µ(Ei) < µ (A) + ε. Note that [ [ A = A ∩ ( Ei) = (A ∩ Ei). i i 7 6 - a b Using consequently σ-semiadditivity and monotonicity of µ, one obtains: X X ∗ µ(A) 6 µ(A ∩ Ei) 6 µ(Ei) < µ (A) + ε.