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Measure Theory and Lebesgue Integration

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Measure Theory and Lebesgue Integration Appendix D Measure theory and Lebesgue integration ”As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.” Albert Einstein (1879-1955) In this appendix, we will briefly recall the main concepts and results about the measure theory and Lebesgue integration. Then, we will introduce the Lp spaces that prove especially useful in the analysis of the solutions of partial differential equations. Contents D.1 Measure theory . 107 D.2 Riemann integration . 109 D.3 Lebesgue integration . 111 D.4 Lp spaces . 116 D.1 Measure theory Measure theory initially was proposed to provide an analysis of and generalize notions such as length, area and volume (not strictly related to physical sizes) of subsets of Euclidean spaces. The approach to measure and integration is axomatic, i.e. a measure is any function µ defined on subsets which satisfy a cetain list of properties. In this respect, measure theory is a branch of real analysis which investigates, among other concepts, measurable functions and integrals. D.1.1 Definitions and properties Definition D.1 A collection S of subsets of a set X is said to be a topology in X is S has the following three properties: (i) S and X S, ∅ ∈ ∈ 107 108 Chapter D. Measure theory and Lebesgue integration (ii) if V S for i = 1, . , n then, V V V S, i ∈ 1 ∩ 2 ∩ · · · ∩ n ∈ (iii) if V is an arbitrary collection of members of S (finite, countable or not), then V inS. { α} α α if S is a topology in X, then X is called a topological space and the members of S are!called the open sets in X. Definition D.2 A collection F of subsets of a set X is said to be a σ-algebra in X if F has the following three properties: (i) X F , ∈ (ii) if A F , then Ac F , where Ac is the complement of A relatie to X, ∈ ∈ (iii) if A = ∞ A and if A F for all n, then A F . n=1 n n ∈ ∈ If F is a σ-algebra! in X, then the pair (X, F ) is called a measurable space . If X is a measurable space, Y is a topological space and f is a mapping of X into Y , then f is said to be measurable provided that 1 f − (V ) is a measurable set in X for every open set V in Y . Definition D.3 A measure is a function defined on a σ-algebra F over a set X and taking values in the interval [0, [ such that the following properties are satisfied: ∞ (i) the emptyset has measure zero, µ( ) = 0; ∅ (ii) countable additivity: if (Ei) is a countable sequence of pairwise disjoint sets in F , then ∞ ∞ µ Ei = µ(Ei) . " $ i#=1 %i=1 The triple (X, F, µ) is then called a measure space and the members of F are called measurable sets. For measure spaces that are also topological spaces, various compatibility conditions can be placed for the measure and the topology. Theorem D.1 Let f and g be real measurable functions on a measurable space X, let Φ be a continuous mapping of the plane into a topological space Y and define h(x) = Φ(u(x), v(x)) for x X. Then, ∈ h : X Y is measurable. → Proposition D.1 If E is a measurable set in X and if 1 if x E χ (x) = ∈ E 0 if x / E & ∈ then, χE is a measurable function called the characteristic function of the set E. Definition D.4 Let (a ) be a sequence in [ , ] and put n −∞ ∞ b = sup a , a , a , . k = 1, 2, 3, . k { k k+1 k+2 } and β = inf b , b , b , . We call β the upper limit of (a ) and write { 1 2 3 } n β = lim sup an . n →∞ D.2. Riemann integration 109 The lower limit is defined analogously by interchanging sup and inf in the previous definition. Moreover, we have lim inf an = lim sup( an) , n − n − →∞ →∞ and, if (an) converges, then lim sup an = lim inf an = lim an . n n n →∞ →∞ → Supose (fn) is a sequence of real functions on a set X. Then supn fn and lim supn fn are the functions defined on X by: →∞ (sup fn)(x) = sup(fn(x)) , (lim sup fn)(x) = lim sup(fn(x)) . n n n n →∞ →∞ Moreover, if f(x) = limn fn(x) the limit being assumed to exist at every x X, then f is called the →∞ ∈ pointwise limit of the sequence (fn). Theorem D.2 If fn : X R is measurable, for n = 1, 2, . and → g = sup fn , h = lim sup fn , n 1 n ≥ →∞ then g and h are both measurable. Corollary D.1 The limit of very pointwise convergent sequence of complex measurable functions is mea- surable. If f and g are measurable then so are max(f, g) and min(f, g). In particular, this is true of the functions + f = max(f, 0) and f − = min(f, 0) , − respectively called the positiveand negative parts of f. We have f = f + f and f = f + + f . − − | | − Proposition D.2 If f = g h, g 0 and h 0 then f + g and f h. − ≥ ≥ ≤ − ≤ D.1.2 Completeness A measurable set X is called a null set if µ(X) = 0. By extension, a subset of a null set is called a negligible set. A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is called complete if every negligible set is measurable. A measure can be extended to a complete one by considering the σ-algebra of subsets Y which differ by a negligible set from a measurable set X, that is, such that the symmetric difference of X and Y is contained in a null set. One defines µ(Y ) to equal µ(X). D.1.3 Non-measureable sets D.2 Riemann integration Suppose that a function f is bounded on the interval [a, b], where a, b R and a < b and consider a ∈ dissection ∆ : a = x < x < < x = b of [a, b]. Then, 0 1 · · · n 110 Chapter D. Measure theory and Lebesgue integration Definition D.5 (Riemann sum) The lower Riemann sum of f(x) corresponding to the dissection ∆ is defined as the following sum: n s(f, ∆) = (xj xj 1 inf f(x) − − x [xj 1,xj ] %j=1 ∈ − anf the upper Riemann sum of f(x) corresponding to the dissection ∆ is given by the sum: n S(f, ∆) = (xj xj 1 sup f(x) . − − x [xj 1,xj ] %j=1 ∈ − Theorem D.3 Suppose that a function f : R R is bounded on [a, b], where a < b, a, b R and that → ∈ ∆ and ∆ are two dissections of [a, b] such that ∆ ∆. Then, & & ⊆ s(f, δ&) s(f, ∆) and S(f, ∆) S(f, ∆&) . ≤ ≤ If ∆&& is another dissection of [a, b] then, s(f, ∆&) S(f, ∆&&) . ≤ Definition D.6 For all dissections ∆ of [a, b], the real number I−(f, a, b) = sup s(f, ∆) is called the ∆ lower integral of f(x) over [a, b] and the real number I+(f, a, b) = inf S(f, ∆) is called the upper integral ∆ of f(x) over [a, b]. Theorem D.4 Suppose that a function f is bounded on the interval [a, b], where a < b and a, b R. ∈ Then + I−(f, a, b) I (f, a, b) . ≤ Definition D.7 Suppose that a function f is bounded on the interval [a, b], where a < b and a, b R + ∈ and suppose that I−(f, a, b) = I (f, a, b). Then, the function f is said to be Riemann integrable over [a, b] and we write b + f(x) dx = I−(f, a, b) = I (f, a, b) . 'a Lemma D.1 Suppose that a function f is bounded on the interval [a, b], where a < b and a, b R. The ∈ following two statements are equivalent: (i) f is Riemann integrable over [a, b] (ii) given any ε > 0, there exists a dissection ∆ of [a, b] such that S(f, ∆) s(f, ∆) < ε . − We have several additional properties of Riemann integrals. Lemma D.2 Suppose that f, g are Riemann integrable over [a, b], where a < b and a, b R. Then ∈ b b b (i) f + g is Riemann integrable over [a, b] and (f(x) + g(x)) dx = f(x) dx + g(x) dx, 'a 'a 'a b b (ii) for every c R, cf is Riemann integrable over [a, b] and cf(x) dx = c f(x) dx, ∈ 'a 'A D.3. Lebesgue integration 111 b (iii) if f(x) 0 for every x [a, b] then f(x) dx 0, ≥ ∈ ≥ 'a b b (iv) if f(x) g(x) for every x [a, b] then f(x) dx g(x) dx. ≥ ∈ ≥ 'a 'a Proposition D.3 Suppose that f is Riemann integrable over [a, b] where a, b R and a < b. Then, ∈ for every real number c [a, b], f is Riemann integrable over [a, c] and Riemann integrable over [c, b]. ∈ Moreover, we can write: b c b f(x) dx = f(x) dx + f(x) dx . 'A 'a 'c Similarly, we have the following result. Proposition D.4 Suppose that a, b, c R and that a < c < b. Suppose further that f is Riemann ∈ integrable over [a, c] and is Riemann integrable over [c, b].
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