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Measure and Integration

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Measure and Integration Appendix Measure and Integration In this appendix we collect some facts from measure theory and integration which are used in the book. We recall the basic definitions of measures and integrals and give the central theorems of Lebesgue integration theory. Proofs may be found for instance in [Rud87]. A.1 Measurable Functions and Integration Let X be a set. A σ -algebra on X is a set A, whose elements are subsets of X, such that • the empty set lies in A and if A ∈ A, then its complement X A lies in A, • the set A is closed under countable unions. It follows that a σ -algebra is closed under countable sections and that with A,B the set A B lies in A. The set P(X) of all subsets of X is a σ -algebra and the intersection of arbitrary many σ -algebras is a σ -algebra. This implies that for an arbitrary set S ⊂ P(X) there exists a smallest σ -algebra A containing S. In this case we say that S gen- erates A. For a topological space X,theσ -algebra B = B(X) generated by the topology of X, is called the Borel σ -algebra of X. The elements of B(X) are called Borel sets.IfA is a σ -algebra on X, then the pair (X, A) is called a measurable space. The elements of A are called measurable sets. Amapf : X → Y between two measurable spaces is called a measurable map −1 if the preimage f (A) is measurable for every measurable set A ∈ AY . The com- position of two measurable maps is measurable. We equip the real line R and the complex plane C with its respective Borel σ - algebra. Lemma A.1.1 Let (X, A) be a measurable space. (a) A function f : X → R is measurable if and only if for every a ∈ R the set f −1((a, ∞)) is in A. A. Deitmar, Automorphic Forms,Universitext, 241 DOI 10.1007/978-1-4471-4435-9, © Springer-Verlag London 2013 242 Appendix Measure and Integration (b) A function f : X → C is measurable if and only if Re f and Im f are measur- able. (c) If f,g : X → C are measurable, then so are f + g,f · g, and |f |p for p>0. (d) If f,g : X → R are measurable, then so are max(f, g) and min(f, g). (e) If a sequence of measurable functions fn : X → C converges point-wise to a function f : X → C, then f is measurable as well. In the following it is helpful to consider functions with values in the interval [0, ∞], where we equip [0, ∞] with the obvious topology and the correspond- ing Borel Σ-algebra. A function f : X →[0, ∞] is measurable if and only if f −1((a, ∞]) ∈ A holds for every a ∈ R. The assertions (c), (d) and (e) of the lemma remain valid for functions f : X →[0, ∞]. A measure μ on a measurable space (X, A) is a map μ : A →[0, ∞] such that μ(∅) = 0 and • ∞ = ∞ μ( n=1 An) n=1 μ(An) holds for every sequence (An)n∈N of pairwise dis- joint sets An ∈ A. It is easy to deduce the following. • μ(A ∪ B) = μ(A) + μ(B) − μ(A ∩ B) for all A,B ∈ A. • A ⊆ ∈ N For a sequence (An)n∈N in with An An+1 for all n , the sequence μ(An) = ∞ converges to μ(A), where A n=1 An. • A ⊇ ∈ N ∞ For a sequence (An)n∈N in with An An+1 for all n and μ(A1)< ,the = ∞ sequence μ(An) converges to μ(A), where A n=1 An. Let μ : A →[0, ∞] be a measure on (X, A). The triple (X, A,μ) is called a measure space. Let (X, A,μ) be a measure space. A step function is a measurable function s : X →[0, ∞] which takes only finitely many values. Any such function is of the form = m ∈ A s i=1 ai1Ai with pairwise disjoint Ai . For such a step function we define its integral as m def sdμ= aiμ(Ai) ∈[0, ∞]. X i=1 For a measurable function f : X →[0, ∞] we define fdμ= sup sdμ: 0 ≤ s ≤ f ; s is a step function . X X The function is called integrable if fdμ<∞. A measurable function f : X → X + R is called integrable if |f | is integrable. In that case the functions f = max(f, 0) − =− = + − and f min(f, 0) are both integrable and we set X fdμ X f dμ − = + X f dμ. A complex valued function f u iv is called integrable if its real and = + imaginary parts u, v are. In that case one defines X fdμ X udμ i X vdμ. A.2 Fubini’s Theorem 243 Proposition A.1.2 Let (X, A,μ) be a measure space. A measurable function f : X → C is integrable if and only if its absolute value |f | has finite integral. In that case one has def fdμ ≤ f 1 = |f | dμ. X X The following two theorems are of central importance. Theorem A.1.3 (Monotone convergence theorem) Let (fn)n∈N be a point-wise monotonically increasing sequence of measurable functions ≥ 0. For x ∈ X set f(x)= limn fn(x) ∈[0, ∞]. Then one has fdμ= lim fn dμ. X n X Theorem A.1.4 (Dominated convergence theorem) Let (fn)n∈N be a sequence of complex valued integrable functions, which converges point-wise to a func- tion f . Suppose there exists an integrable function g such that |fn|≤|g| holds for every n ∈ N. Then f is integrable and one has fdμ= lim fn dμ. X n X A.2 Fubini’s Theorem A measure μ on a measurable space (X, A) is called a σ -finite measure if there are ⊂ ∈ N = ∞ ∞ countably many subsets Xj X, j with X j=1 Xj and μ(Xj )< for every j ∈ N. Examples A.2.1 • The Lebesgue measure on X = R is σ -finite, since R can be written as a countable union of the intervals [k,k + 1] with k ∈ Z. • The counting measure is not σ -finite on X = R, since R is uncountable. For two σ -finite spaces (X, A,μ) and (Y, C,ν) one shows that there exists a unique measure μ · ν on the σ -algebra A ⊗ C, which is generated by all sets of the form {A × C : A ∈ A,C∈ C}, such that μ · ν(A × C) = μ(A)ν(C), A ∈ A,C∈ C. The measure μ · ν is called the product measure of μ and ν. 244 Appendix Measure and Integration Theorem A.2.2 (Fubini’s theorem) Let (X, μ) and (Y, ν) be σ -finite measure spaces and let f be a measurable function on X × Y . ≥ (a) If f 0, then the partial integrals X f(x,y)dμ(x) and Y f(x,y)dν(y) define measurable functions and one has the Fubini formula, f(x,y)dμ · ν(x,y) = f(x,y)dν(y)dμ(x) × X Y X Y = f(x,y)dμ(x)dν(y). Y X (b) If f is complex valued and if one of the iterated integrals f(x,y) dν(y)dμ(x) or f(x,y) dμ(x)dν(y) X Y Y X is finite, then f is integrable with respect to the product measure and the Fubini formula holds. In this book, we use the Fubini theorem for Haar measures only. All Haar mea- sures occurring in this book are σ -finite. But as we did not mention this explicitly each time, we will also give a version of Fubini’s theorem for Radon measures which works without the σ -finiteness condition (see [DE09], Appendix). Theorem A.2.3 (Theorem of Fubini for Radon measures) Let μ and ν be Radon measures on the Borel sets of locally compact spaces X and Y , respec- tively. Then there exists a unique Radon measure μ · ν on X × Y such that 1. If f : X × Y → C is μ · ν-integrable, then the partial integrals X f(x,y)dx and Y f(x,y)dy define integrable functions such that Fu- bini’s formula holds: f(x,y)d(x,y) = f(x,y)dy dx = f(x,y)dx dy. X×Y X Y Y X 2. If f is measurable such that A ={(x, y) ∈ X × Y : f(x,y) = 0} is σ -finite, and if one of the iterated integrals f(x,y) dy dx or f(x,y) dxdy X Y Y X is finite, then f is integrable and the Fubini formula holds. A.3 Lp-Spaces 245 A.3 Lp-Spaces Let (X, A,μ) be a measure space. For 1 ≤ p<∞ write Lp(X) forthesetofall measurable functions f : X → C such that 1 p def p f p = |f | dμ < ∞. X A function in L1(X) is called integrable, as we already know. A function in L2(X) is called square integrable. Further, let L∞(X) be the set of all measurable functions f : X → C for which there exists a set N of measure zero such that f is bounded on the complement X N. Then def f ∞ = inf 0 <c≤∞:∃set of measure zero N with f(X N) ≤ c is a semi-norm on the complex vector space L∞(X). Proposition A.3.1 (Minkowski inequality) Let p ∈[1, ∞]. For all f,g ∈ Lp(X) one has f + g ∈ Lp(X) with f + g p ≤ f p + g p.
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