Preparatory Material for the European Intensive Program in Bydgoszcz 2011 Analytical and computer assisted methods in mathematical models September 4{18 Basics on the Lebesgue integral and the divergence theorem Wolfgang Reichel KIT Universität des Landes Baden-Württemberg und nationales Forschungszentrum in der Helmholtz-Gesellschaft www.kit.edu
Ref.: W. Rudin, Real and Complex Analysis, 3 rd Ed., McGraw-Hill, 1987, Chapter 1 3. Let be a set, e.g. = R n, and let P() = be the set of all subsets of. Definition 1 (σ-algebra) A system of sets M P() is called a σ-algebra over if (i) M (ii) A M = \ A M (iii) A i M i N = A i M Definition 2 (positive measure) Let M be σ-algebra over. A map µ : M [0, ] is called a positive measure, if ( ) A i M i N with A i A j = for i j = µ A i = µ(a i ). Let I = I 1 I n = (a 1, b 1 ) (a n, b n ) be an open intervall in R n. Then the volume of I is defined by I = (b 1 a 1 )... (b n a n ). The same applies if one or more of the component-intervals (a i, b i ) are replaced by closed, semi-closed, semi-open intervals. Definition 3 (outer measure on R n ) Let A R n be an arbitrary set. Then { } λ(a) := inf I i : A I i und I i bounded interval i N is called the outer measure of the set A. Remark: λ is not a positive measure on P(R n ). Definition 4 (Lebesgue σ-algebra, Caratheodory) A set A R n is called Lebesgue-measurable (short: A L(R n )) if λ(e) = λ(a E) + λ(a c E) E R n. If R n is Lebesgue-measurable, then let L() = {A : A is Lebesgue-measurable}. Theorem 5 L(R n ) is a σ-algebra. The outer measure λ (see Definition 3) is invariant under Eulcidean motions and if it is restricted to L(R n ) then it becomes a positive, complete measure on L(R n ). In the following let R n be a Lebesgue-measurable set. For f : R = [, ] let f + = max{f, 0}, f = min{f, 0}. Hence f = f + f.
Definition 6 (mesasurable functions) (i) A function f : R is called measurable, if f 1( (α, ] ) L() for all α R, (ii) A function s : R is called an elementary function, if s possesses only finitely many values α 1,..., α k. In this case s = k α i χ Ai, A i = s 1 (α i ). Definition 7 (Lebesgue-integral for non-negative functions) (i) Let s = k α iχ Ai be a measurable elementary function. Then s dx := is called the Lebesgue-integral of s over. k α i λ(a i ) (ii) Let f : [0, ] be measurable. Then f dx := sup s dx, S = {s : R measurable elementary function, 0 s f} s S is called the Lebesgue-integral of f over the set. Definition 8 (Lebesgue-integral for real- or complex-valued functions) Let IK = R or C. L 1 () := {f : IK measurable : f dx < }. For f L 1 () let f 1 = Rf, f 2 = If. Then f dx := dx f 1 + f1 is called the Lebesgue-integral of f over the set. ( dx + i f 2 + dx Definition 9 (f = g a.e.) Let f, g : IK be measurable. Then we say f = g almost everywhere, if there exists a set N of measure 0 such that f(x) = g(x) x \ N. Equality almost everywhere is an equivalence relation. f2 Definition 10 (essential supremum) Let F : R be measurable. Then is called the essential supremum of F. ess sup F := inf{s R : F (x) s a.e. in }. ) dx
Definition 11 (The space L p ()) (a) For 1 p < let L p () = {u : R measurable: u p dx < }. (b) For p = let L () = {u : R measurable: ess sup u < }, where ess sup v = inf{s R : v(x) s for almost all x }. Definition 12 (Norm on L p ()) For 1 p < let and ( u p := ) 1/p u p dx u := ess sup u. Then (L p (), p ) is a Banachspace. L 2 () is a Hilbertspace with inner product f, g := fg dx. Theorem 13 (Minkowski and Hölder inequalities) (i) u + v p u p + v p for all u, v L p (). (ii) Let 1 p, q such that 1 + 1 = 1. Then p q uv dx u p v q for all u L p () and all v L q (). Theorem 14 (Dual space of L p ()) Let 1 p < and let φ : L p () R be a continuous linear functional. Then there exists a unique v L q (), 1 + 1 = 1, such that p q For short: (L p ()) = L q (). φ(u) = uv dx for all u L p (). Note: In general the theorem fails for p =, i.e., (L ()) L 1 (). Theorem 15 Let 1 p and u L p (). If (u k ) k N is a sequence of functions in L p () such that lim k u k u p = 0 then there exists a subsequence (u kl ) l N such that lim u k l (x) = u(x) for almost all x. l
Theorem 16 Let 1 p <. The set C c () of continuous functions with compact support in are dense in L p (). Here the support of a function is defined as supp f = {x : f(x) 0}. Definition 17 (Locally integrable functions) Let 1 p. Define L p loc () = {u : R measurable s.t. u Lp (K) for every compact set K }. L p loc () is a vector space. Theorem 18 (Monotone convergence) Let (u k ) k N be a sequence of measurable functions on such that 0 u 1 u 2 u 3.... Then u(x) := lim k u k (x) exists for almost all x and u k dx = u dx. lim k Theorem 19 (Dominated convergence) Let (u k ) k N be a sequence of measurable functions on. If there exists w L 1 () such that u k (x) w(x) for almost all x and all k N and if u(x) := lim k u k (x) exists almost everywhere in then lim k u k dx = u dx. Theorem 20 (Fatou s Lemma) Let (u k ) k N be a sequence of measurable functions on such that u k (x) 0 almost everywhere on. Then lim inf u k dx lim inf u k dx. k N k N Definition 21 An open, connected set R n is called a C 1 -domain (a Lipschitz-domain) if for every x 0 there exists a radius r > 0 and a C 1 -function (Lipschitz-function) γ : R n 1 R such that (up to rotation of the coordinate frame) Note that as a consequence B r (x 0 ) = {x = (x, x n ) B r (x 0 ) : x n > γ(x )}. B r (x 0 ) = {x = (x, x n ) B r (x 0 ) : x n = γ(x )}. For x = (x, γ(x )) B r (x 0 ) the vector ν(x) = ( γ(x ), 1) 1 + γ(x ) 2 is called the exterior unit-normal of at x. In case of a Lipschitz-domain the exterior unitnormal ν exists a.e. with respect to the surface-measure on.
Theorem 22 (Divergence theorem, Gauss theorem) Let R n be a bounded C 1 -domain and let ν be the exterior unit-normal on. Then f dx = fν i dσ x i for every function f C 1 (). The result also holds if is a bounded Lipschitz-domain. Often the divergence theorem appears in the following form div F dx = F ν dσ for a vectorfield F : R n. The components of F = (F 1,..., F n ) have to satisfy F i C 1 () for i = 1,..., n. Lemma 23 (Green s identities) Let u, v C 2 (). Then u dx = u ν dσ, u v dx = u v dx + u v ν dσ, (u v v u) dx = (u v v u) ν dσ.