AN INTRODUCTION TO GEOMETRIC MEASURE THEORY AND AN APPLICATION TO MINIMAL SURFACES ( DRAFT DOCUMENT) Academic Year 2016/17 Francesco Serra Cassano

AN INTRODUCTION TO GEOMETRIC MEASURE THEORY AND AN APPLICATION TO MINIMAL SURFACES ( DRAFT DOCUMENT) Academic Year 2016/17 Francesco Serra Cassano Contents I. Recalls and complements of measure theory. I.1 Measures and outer measures, approximation of measures I.2 Convergence and approximation of measurable functions: Severini- Egoroff s theorem and Lusin s theorem. I.3 Absolutely continuous and singular measures. Radon-Nikodym and Lebesgue decomposition theorems. I.4 Signed vector measures: Lebesgue decomposition theorem and polar decomposition for vector measures. I.5 Spaces L p (, µ) and their main properties. Riesz representation theorem. I.6 Operations on measures. I.7 Weak*-convergence of measures. Regularization of Radon measures on R n. II. Differentiation of Radon measures. II.1 Covering theorems and Vitali-type covering property for measures on R n. II.2 Derivatives of Radon measures on R n. Lebesgue-Besicovitch differentiation theorem for Radon measures on R n. II.3 Extensions to metric spaces. III. An introduction to Hausdorff measures, area and coarea formulas. III.1 Carathéodory s construction and definition of Hausdorff measures on a metric space and their elementary properties; Hausdorff dimension. III.2 Recalls of some fundamental results on Lipschitz functions between Euclidean spaces and relationships with Hausdorff measures. III.3 Hausdorff measures in the Euclidean spaces; H 1 and the classical notion of length in R n ; isodiametric inequality and identity H n = L n on R n ; k-dimensional densities. III.4 Area and coarea formulas in R n and some applications. III.5 Extensions to metric spaces. IV. Rectifiable sets and blow-ups of Radon measures. IV.1 Rectifiable sets of R n and their decomposition in Lipschitz images. IV.2 Approximate tangent planes to rectifiable sets. IV.3 Blow-ups of Radon measures on R n and rectifiability. IV.4 Extensions to metric spaces. V. An introduction to minimal surfaces and sets of finite perimeter. V.1 Plateau problem: nonparametric minimal surfaces in R n, area functional and its minimizers. V.2 Direct methods of the calculus of variations and application to the existence of minimizers for the Plateau problem. V.3 Sets of finite perimeter, space of bounded variation functions and their main properties; sets of minimal boundary. 1

2 V.4 Structure of sets of finite perimeter and reduced boundary. V.5 Regularity of minimal boundaries. V.6 Extensions to metric spaces. SOME BASIC NOTATION If A, B are sets then the symmetric difference between A and B will be denoted by A B := (A \ B) (B \ A). We shall tipically work in a metric space with a metric d, although we will present some notions and results in more general settings. In some chapters however we mainly deal with the Euclidean n- space R n. Here the basic notation used in metric spaces throughout these notes. The closed and open balls with centre x and radius r, 0 < r <, are denoted by In R n we also set B(x, r) = {y : d(x, y) r}, U(x, r) = {y : d(x, y) < r}. B(r) = B(0, r), U(r) = U(0, r), S(x, r) = B(x, r) and S(r) = B(0, r); If B = B(x, r) (respectively B = U(x, r)) and α > 0, we denote αb = B(x, α r) (respectively αb = U(x, α r)). When α = 5 we will call 5B an enlargement of B and we will denote it by ˆB. The diameter of a nonempty subset A is d(a) = diam(a) = sup {d(x, y) : x, y A}. We agree d( ) = 0. If x and A and B are non-empty subsets of, the distance from x to A and the distance between A and B are, respectively, d(x, A) = inf {d(x, y) : y A}, d(a, B) = inf {d(x, y) : x A, y B}. For ɛ > 0 the open ɛ-neighbourhood of A is If A R n, then I ɛ (A) = {x : d(x, A) < ɛ}. A = L n (A) where L n denote the n-dimensional Lebesgue outer measure. 1. Recalls and complements of measure theory ([AFP, GZ, Mag, R1, SC]). Motivation: The main goal is to recall and complement some notions and results of measure theory such as: outer measure, measure, signed measures and vector measure with their properties and relationships; measurable functions and their properties; L p spaces and Riesz representation theorem; convergence of measures.

1.1. Measures and outer measures, approximation of measures. Measures and outer measures. Let us quickly recall some important notions and results of abstract measure theory (see [GZ]). Tipically there are two approachs in abstract measure theory: one by using measure, may be more ordinary in the literature, and one by outer measure due to Carathéodory. Firstly let us introduce the so-called set-theoretic approach where we introduce the notion of measure and measurable set, only assuming that the environment is a set. Definition 1.1. Let denote a set and P() denote the class of all subsets of. (i) A set function ϕ : P() [0, ] is called an outer measure (o.m.) on if (OM1) ϕ( ) = 0, (OM2) ϕ(a) ϕ(b) if A B (monotonicity), (OM3) ϕ( i=1a i ) ϕ(a i ) for any sequence (A i ) i (countable subadditivity). i=1 (ii) A set E is called ϕ-measurable (with respect to an o.m. ϕ on ), if ϕ(a) = ϕ(a E) + ϕ(a \ E) A. The class of ϕ-measurable sets will be denoted by M ϕ. (iii) A σ-algebra M on is a (nonempty) class of subsets M P() satisfying the two following properties: (σa1) if E M, then \ E M ; (σa2) for each sequence (E i ) i M, then i=1 E i M. (iv) A measure µ on is a set function µ : M [0, ], where M is a σ-algebra on, satisfying the following two properties: (M1) µ( ) = 0 ; (M2) µ( i=1e i ) = µ(e i ) i=1 (countable additivity) for each disjoint sequence (E i ) i M. The structure composed by the triple (, M, µ), or also the couple (, M), is called measure space and the sets contained in M are called measurable sets. (v) Let (, M, µ) be a measure space. The measure µ is said to be finite, if µ() < ; it is said to be σ-finite, if there exists a sequence ( i ) i M such that = i=1 i and µ( i ) < for each i. (v) Let (, M, µ) be a measure space. A point x M is said to be an atom if the singleton {x} M and µ({x}) > 0. The set of atoms of µ will be denoted by S µ and µ is said to be atomic if S µ. If µ is finite or σ-finite the set of atoms S µ is at most countable. 3

4 (vii) A function f : R := [, ] is called measurable with respect to an o.m. ϕ (respectively with respect to a measure µ : M [0, ]) if f 1 (U) is ϕ-measurable (respectively f 1 (U) M) for each open set U in R. (viii) A simple function s : R is one that assumes only a finite number of values. More precisely, s is a simple function if and only if it can be represented as s(x) = k a i χ Ai (x) x, i=1 with a i R, A i (i = 1,..., k), = k i=1a i and A i A j = if i j. Based on the ideas of H. Lebesgue, it is well known that a theory of an abstract integration can be carried out on a general measure space (, M, µ) and we refer to [GZ, Chap. 6] or [R1, Chap. 1] for its complete treatment. In particular, let us recall that, given a measurable function f : [, ], it is possible to make a sense to the value integral of f with respect to µ, denoted f dµ [, ]. When the integral is finite, that is f dµ (, ), f is said to be integrable or also summable. Example 1.2. Let (, M) be a measure space, then we define the following set functions on M, which turns out to be measures, as it can easily be proved. (i) (counting measure) We define the set function # : M [0, ], #( ) := 0, #(E) as the cardinality of E if it is finite, #(E) = otherwise. (ii) (Dirac measures) With each x we associate the set function δ x : M [0, ] defined by δ x (E) := 1 if x E, δ x (E) := 0 otherwise. If (x h ) h and (c h ) h [0, ) is a sequence such that the series h=1 c h is convergent, we can define the set function h=1 c h δ xh : M [0, ] ( ) c h δ xh (E) := c h. h=1 {h: x h E} Measures of this kind are called purely atomic. Now, we enrich the environment, by adding a topology and we require compatibility between topology and measure. Definition 1.3. Let (, τ) denote a topological space and denote B() the σ-algebra of Borel sets of, i.e. the smallest σ-algebra of which contains the open and closed sets of. (i) An o.m. ϕ on is called a Borel o.m. if the class of ϕ-measurable sets M ϕ B(). (ii) An o.m. ϕ on is called a Borel regular o.m. if it is a Borel o.m. and for each A there exists B B() with B A and ϕ(a) = ϕ(b). (iii) An o.m. ϕ on is called a Radon o.m. if it is a Borel regular o.m. and ϕ(k) < for each compact set K.

(iv) An o.m. ϕ on a metric space (, d) is called a Carathéodory o.m. (or also a metric o.m.) if ϕ(a B) = ϕ(a) + ϕ(b) A, B with d(a, B) > 0, where d(a, B) := inf{d(a, b) : a A, b B}. (v) A measure µ : M [0, ] on is called a Borel measure if M = B(). (vi ) A measure µ : M [0, ] on is called a Radon measure if it is a Borel measure and µ(k) < for each compact set K. (vii ) An o.m ϕ on (respectively a Borel measure µ : B() [0, ]) is said to locally finite if for each x there exist an open neiborghood U x of x such that ϕ(u x ) < (respectively µ(u x ) < ). Remark 1.4. We stress that the notion of Radon o.m. (respectively Radon measure) in a general topological space (, τ) may actually differ in the current literature from one given in Definition 1.3 (iii) (respectively Definition 1.3 (vi)). Indeed it could be requested that ϕ (respectively µ) must satisfy to be finite on compact sets and approximation properties (i) and (ii) of Theorem 1.15. The two notions agree on a separable, locally compact metric space (, d) because of Theorem 1.15. The following basic properties of outer measures are well known (see [GZ]). Theorem 1.5. (i) Let ϕ be an o.m. on. Then the class of ϕ-measurable sets M ϕ is a σ-algebra on and ϕ : M ϕ [0, ] is a measure. (ii) If ϕ(n) = 0, then N M ϕ. (iii) (Carathéodory s criterion) Let ϕ be a Carathéodory o.m. on a metric space (, d). Then ϕ is a Borel o.m. Remark 1.6. The property of Theorem 1.5 (ii) is characteristic to an outer measure but is not enjoyed by measures. A measure µ : M [0, ] with the property that all subsets of sets of µ-measure zero are measurable, is said to be complete and (, M, µ) is called complete measure space. Not all measures are complete, but this is not a crucial defect since every measure can easily be completed by enlarging its domain of definition to include all subsets of measure zero, that is by replacing M with its completion denoted M (see [R1, Theorem 1.36] or [GZ, Theorem 4.45]). Example 1.7. (i) Let L n denote the Lebesgue o.m. on R n. Then L n is a Radon measure on R n. The class M n M L n is called the class of Lebesgue measurable sets. (ii) Let s [0, ) and H s denote the s-dimensional Hausdorff measure on R n. Then H s is a Borel regular o.m. on R n, but it is not a Radon o.m. unless s n. We will deeply study these measures in Chapter III. (iii) Let A be a non Borel set of R n (why does A exists?); let ϕ : P(R n ) [0, + ] be the set function defined as { 0 if E A ϕ(e) := + if E \ A. It is easy to see, by the definition, that ϕ is a Carathéodory outer measure on R n. However it is not Borel regular because it does not exist a Borel set B A such that ϕ(b) = ϕ(a). 5

6 (iv) Let ϕ : P(R n ) [0, + ] be the set function defined as { 0 if E = ϕ(e) :=. 1 if E It is easy to see, by the definition, that ϕ is an o.m. on R n and M ϕ = {, R n }. In particular, it is not a Borel o. m. By Theorem 1.5 (i) we see that to every o.m. ϕ on is associated the measure space (, M ϕ, ϕ) Question: Given a measure space (, M, µ) is there an associated o.m. µ : P() [0, ] such that µ = µ on M? There is a simple procedure due to Carathéodory to generate from a measure µ : M [0, ] an outer measure µ. Moreover µ is also unique (see [GZ, Theorems 4.47 and 4.48]). Theorem 1.8 (Carathéodory-Hahn extension theorem). Let (, M, µ) be a measure space, let µ (E) := inf {µ(a) : A E, A M} for each E. Then (i) µ is an o.m.; (ii) µ (A) = µ(a) whenever A M; (iii) M M µ ; (iv) Let N be a σ-algebra with M N M µ and suppose that ν is a measure on N such that ν = µ on M. Then ν = µ on N, provided that µ is σ-finite. µ is called the o.m. generated by µ. Let us recall three important results on approximation of measures by open and closed sets. The first result is also contained in [GZ, Theorem 4.17]). Theorem 1.9 (Approximation of outer measures by open and closed sets). Let ϕ be a Borel (respectively a Borel regular) o.m. on a metric space (, d) and let B be a Borel set (respectively a ϕ-measurable set). (i) Suppose that ϕ(b) <, then for each ɛ > 0 there exists a closed set F B such that ϕ(b \ F ) < ɛ. (ii) Suppose B i=1v i where each V i is an open set with ϕ(v i ) <. Then there is an open set U B such that ϕ(u \ B) < ɛ. Remark 1.10. Notice that, if ϕ(b) =, then the conclusion of Theorem 1.9 (i) may fail. For instance, consider = R, ϕ = #, B = (0, + ). Then, for each closed set F (0, + ), #(B \ F ) =. The conclusion of Theorem 1.9 (ii) may also fail by means of the same example if the assumptions are dropped. Remark 1.11. It is easy to that the conclusion of Theorem 1.9 still hold when replacing outer measure ϕ with a Borel measure µ.

The first important consequence of Theorem 1.9 is the following Corollary 1.12. Let ϕ be a Borel (respectively a Borel regular) o.m. on a metric space (, d). Suppose there exists a sequence of open sets (V i ) i such that ( ) = i=1v i with ϕ(v i ) < i. Then for each B B() (respectively B M ϕ ) (i) ϕ(b) = inf{ϕ(u) : U B, U open}; (ii) ϕ(b) = sup{ϕ(c) : C B, C closed}. Proof. See, for instance, [SC, Corollary 1.19]. The second important consequence of Corollary 1.12 and the Carathéodory-Hahn extension theorem (Theorem 1.8) is the following approximation result for Borel measures. Corollary 1.13 (Approximation of Borel measures by open and closed sets). Consider a measure space (, B(), µ) where is a metric space and µ is a Borel measure. Suppose that the assumption ( ) of Corollary 1.12 holds replacing ϕ with µ. Then for each B B() (i) µ(b) = inf{µ(u) : U B, U open}; (ii) µ(b) = sup{µ(c) : C B, C closed}. Proof. See, for instance, [SC, Corollary 1.20]. When (, d) is a separable, locally compact metric space and ϕ (respectively µ) is a Radon outer measure (respectively Radon measure) on, the assumption ( ) is satisfied. Thus the conclusions of Corollary 1.12 (respectively Corollary 1.13) hold. Moreover the approximation from below by means of closed sets can be replaced by compact sets. Let us recall Definition 1.14. Let (, τ) be a topological space. (i) (, τ) is said to be separable if it has a countable dense subset. (ii) (, τ) is said to be locally compact if for each x there is an open set O x such that the closure of O, denoted by O, is compact. Theorem 1.15 (Approximation of Radon measures on l.c.s. metric spaces). Let (, d) be a separable, locally compact metric space and ϕ (respectively µ) be a Radon outer measure (respectively Radon measure) on. Then (i) for each B, ϕ(b) = inf{ϕ(u) : U B, U open} (respectively, for each B B(), µ(b) = inf{µ(u) : U B, U open}); (ii) for each B M ϕ, ϕ(b) = sup{ϕ(k) : K B, K compact} (respectively, for each B B(), µ(b) = sup{µ(k) : K B, K compact}). Remark 1.16. An immediate consequence of Theorem 1.15 is that, if two Radon measures on a locally compact metric space (, d) agree on the class open set, then they have to agree on P(). Before the proof of Theorem 1.15 we need the following topological results, whose the former is well known (see, for instance, [Ro, Proposition 7.6]). 7

8 Lemma 1.17. Let (, d) be a separable metric space and let D = {x i : i N} be dense. Then the family of open sets U := {U(x i, q) : i N, q Q (0, )} is a basis for the topology induced on by the distance, where U(x, r) := {y : d(x, y) < r} if x and r > 0. Lemma 1.18. Let (, d) be a separable metric space. (i) Assume that (, d) is also locally compact. Then there exists an increasing sequence of open sets (V i ) i such that (1.1) = i=1v i, V i is compact for each i. (ii) Assume that there exists a locally finite, Borel regular o.m. ϕ (respectively a Borel measure µ) on (, d). Then there exists an increasing sequence of open sets (V i ) i such that (1.2) = i=1v i, ϕ(v i ) < (respectively µ(v i ) < ) for each i. In particular the conclusion of Corollary 1.12 (respectively of Corollary 1.13) holds. Proof of Lemma 1.18. (i) Recall that, by definition, (, d) is locally compact if and only if x r x > 0 such that U(x, r x ) is compact. Let D = {x i : i N} be dense. From Lemma 1.17, the family U is a basis for the topology and let us enumerate U, that is assume that U = {U i : i N}.Thus there exists a set I(x) N for which U(x, r x ) = i I(x) U i. In particular, there exists a choice function α : N satisfying: (1.3) x U α(x) and U α(x) U(x, r x ) U(x, r x ). Let J := α() N and V i := j (J {1,...,i}) U j if i N. Then, by (1.3), (1.1) follows. (ii) Let us prove the conclusion for a locally finite, Borel regular o.m. ϕ, being equal the proof for a locally finite, Borel measure µ. Recall that ϕ is locally finite if x r x > 0 such that ϕ(u(x, r x )) <. With the same notation of point (i), there now exists a choice function α : N satisfying: (1.4) x U α(x) and U α(x) U(x, r x ). In particular (1.5) ϕ(u α(x) ) ϕ(u(x, r x )) < x. Let J := α() N and Then, by (1.4) and (1.5), (1.2) follows V i := j (J {1,...,i}) U j if i N.

Proof of Theorem 1.15. Let us first notice that, without loss of generality, we can assume that B B(). Indeed, if not, since ϕ is a Borel regular o.m., we can replace B by a Borel set B B and ϕ( B) = ϕ(b). By Lemma 1.18 (i), ( ) of Corollary 1.12 is satisfied. Thus claim (i) follows at once from Corollary 1.12 (i) (respectively from Corollary 1.13 (i)). Let us now prove (ii) for a given ϕ Radon o.m. and B M ϕ. Since each compact set is also closed, from Corollary 1.12 (ii), it follows that ϕ(b) = sup{ϕ(c) : C B, C closed} sup{ϕ(k) : K B, K compact}. Thus we have only to prove that (1.6) ϕ(b) = sup{ϕ(c) : C B, C closed} sup{ϕ(k) : K B, K compact} Let C B be a closed set, let (V i ) i be the sequence of open sets in (1.1) and let K i := C ( i j=1v j ). Then (K i ) i is an increasing sequence of compact sets such that Observe now that C := i=1k i. (1.7) ϕ(k i ) sup{ϕ(k) : K B, K compact} for each i, and, by the continuity of o.m. ϕ on increasing sequences of measurable sets, (1.8) lim i ϕ(k i ) = ϕ(c). By (1.7) and (1.8), (1.6) follows. 1.2. Convergence and approximation of measurable functions: Severini- Egoroff s and Lusin s theorems. Theorem 1.19 (Severini-Egoroff). Let (, M, µ) be a measure space with µ finite. Suppose f h : R (h = 1, 2,... ) and f : R are measurable functions that are finite µ-a.e.on. Also, suppose that (f h ) h converges pointwise µ-a.e. to f. Then for each ɛ > 0 there exists a set A M such that µ( \ A) < ɛ and f h f uniformly on A, that is sup f h (x) f(x) 0 as h. x A. Proof. See [GZ, Theorem 5.15]. Remark 1.20. The hypothesis that µ() < is essential in Severini-Egoroff s theorem. Consider the case of Lebesgue measure L 1 on R and define a sequence of functions by f h = χ [h, ), for each positive integer h. Then, lim h f h (x) = 0 for each x R, but (f h ) h does not converge uniformly to 0 on any set A whose complement R\A has finite Lebesgue measure. Indeed, it would follow that R \ A does not contain any half-line [h, ); that is, for each h, there would exist x [h, ) A withf h (x) = 1, thus showing that (f h ) h does not converge uniformly to 0 on A. 9

10 Theorem 1.21 (Approximation by simple functions). Let (, M) be a measure space and let f : [0, + ] be a measurable function. Then there exists a sequence of measurable simple functions s h : [0, + ) (h = 1, 2,... ) satisfying the properties: (i) 0 s 1 s 2... s h... f; (ii) lim h s h (x) = f(x) x. In particular, if f dµ <, then f s h dµ 0. Proof. See [GZ, Theorem 5.24]. Let us now introduce two spaces of continuous functions which play an important role in measure theory. Definition 1.22. Let (, τ) be a topological space. (i) C 0 c() := {f : R : f is continuous and spt(f) is compact in (, τ)} where (1.9) spt(f) := closure {x : f(x) 0}. (ii) C 0 b() := {f : R : f is continuous and bounded} Remark 1.23. Observe that, if a topological space (, τ) is not locally compact, the space C 0 c() could turn out to be meaningless, that is C 0 c() = {0}.Indeed Exercise: Let (, ) be an infinite-dimensional normed vector space. Then C 0 c() = {0}. Lusin s theorem 1.24 (1912, form on locally compact metric spaces). Let µ be a Radon outer measure on a locally compact, separable metric space. Let f : R be a µ-measurable function such that there exists a Borel set A with µ(a) <, f(x) = 0 x \ A and f(x) < µ a.e. x. Then, for each ɛ > 0, there exists g C 0 c() such that Moreover g can be chosen such that Proof. See [R1, Theorem 2.23]. µ ({x : f(x) g(x)}) < ɛ. sup x g(x) sup f(x). x A consequence of Lusin s theorem is the following useful approximation of measurable functions by means of Borel functions. Corollary 1.25. Let µ be a Radon outer measure on a locally compact, separable metric space and let f : R be a µ-measurable function. Then there exist a Borel function g : R such that f = g µ-a.e. on. Proof. See, for instance, [Fe, 2.3.6].

1.3. Absolutely continuous and singular measures. Radon-Nikodym and Lebesgue decomposition theorems. Firstly, let us introduce some definitions and preliminary results. Definition 1.26. Let (, M) be a measure space and let µ, ν : M [0, ] be two measures. (i) The measure ν is said to be absolutely continuous with respect to the measure µ, written ν << µ, if it holds that µ(e) = 0 ν(e) = 0. (ii) The measures ν and µ are said to be mutually singular, written µ ν, if there exists a measurable set E such that ν(e) = µ( \ E) = 0. The following result justifies why the word continuity is used in this context. Theorem 1.27. Let ν be a finite measure and µ a measure on a measure space (, M). Then the following are equivalent: (i) ν << µ; (ii) lim µ(a) 0 ν(a) = 0, that is, for every ε > 0 δ = δ(ε) > 0 such that ν(e) < ε whenever µ(e) < δ. Proof. See [GZ, Theorem 6.33]. Let us now introduce two fundamental results of measure theory: the Radon- Nikodym and Lebesgue s decomposition theorems. Let (, M, µ) be a measure space and w : [0, ] be measurable. Then it is easy to see that µ w is a measure, absolutely continuous w.r.t. µ. The very remarkable fact, content of the Radon-Nikodym theorem, is that essentially each measure ν, absolutely continuous w.r.t. µ is of this form. Theorem 1.28 (Radon-Nikodym). Let ν and µ be two measures on (, M). Suppose that (i) ν and µ are σ-finite, that is, there exists a sequence ( i ) i M such that = i=1 i and ν( i ) < and µ( i ) < for each i. (ii) ν << µ. Then there exists a measurable function w : [0, ] such that ν = µ w on M, that is, (RN) ν(e) = µ w (E) := w dµ E M. Moreover the function w in (RN) is µ-a.e. unique. Definition 1.29. The function w in (RN) is called the Radon-Nikodym derivative of ν with respect to µ and denoted by w = dν dµ. E 11

12 Remark 1.30. Because ν is σ-finite, then w is also σ-integrable with respect to µ, that is, (σi) 0 w dµ < i i, where ( i ) i is the sequence in statement (i). Proof. See See [GZ, Theorem 6.38] and also [SC, Theorem 1.30]. Remark 1.31. We can actually weaken the assumptions of the Radon-Nikodym theorem. Indeed it is sufficient to require that only µ is σ-finite in order that (RN) holds (see, for instance,[ro, Theorem 23, Chap. 11]). When µ is not σ-finite, the Radon- Nikodym theorem fails (see Exercise I.8). For Radon measures on locally compact, separable metric spaces, the Radon- Nikodym theorem has the following simpler and stronger version. Theorem 1.32 (Radon-Nikodym s theorem for Radon measures). If is supposed to be a locally compact, separable metric space and ν and µ are Radon measures on with ν << µ, then (RN) holds and the Radon-Nikodym derivative w := dν is locally dµ integrable on, i.e. w L 1 loc (, µ), where { } L 1 loc(, µ) := f : R : f is measurable, f dµ < for each compact K. Proof. See [SC, Theorem 1.35]. Historical notes: The first version of Radon-Nikodym s theorem is due to H. Lebesgue ([Le]) and to G. Vitali ([Vitali]) when = R in 1904. Radon extended the result when = R n ([Ra]) in 1913. Eventually Nikodym ([Ni]) extended the result to the abstract setting in 1930. A consequence of the Radon-Nikodym theorem is the following Lebesgue decomposition theorem 1.33. Let ν and µ be σ-finite measures on a measure space (, M). Then there is a decomposition of ν such that K ν = ν ac + ν s, where ν ac and ν s are still measures on (, M) with ν ac << µ and ν s µ. The decomposition is unique. Proof. See [GZ, Theorem 6.39] and also [SC, Theorem 1.36]. Exercise I.10 Let consider µ = L 1, ν = δ 0 as measures on the σ- algebra M 1 of Lebesgue { measurable sets in R,where δ 0 denotes the Dirac measure at 0, that is, 1 if 0 E δ 0 (E) :=. Prove that the Lebesgue decomposition of ν with respect 0 if 0 / E. to µ, ν = ν ac + ν s, is given by ν ac 0 and ν s = ν.

1.4. Signed vector measures. Lebesgue decomposition theorem still holds for a more general class of measures. Namely for set functions ν : M R which still verify basic properties of countable additivity. Definition 1.34 (Signed measures). Let (, M) be a measure space. (i) An extended real valued set function ν : M R is a signed measure if it satisfies the following three properties: (SM1) ν assumes at most one of the values +, ; (SM2) ν( ) = 0; (SM3) For each sequence of disjoint sets (E i ) i M, it holds that ν( i=1e i ) = ν(e i ) where the series on the right either converges absolutely or diverges to or +. (ii) A signed measure ν : M R is said to be absolutely continuous with respect to µ : M [0, ], written ν << µ, if ν(e) = 0 whenever µ(e) = 0. (iii) Two signed measure ν, µ : M R are said to be mutually singular, written ν µ, if there is E M such that ν(e) = µ( \ E) = 0. (iv) A signed measure ν : M R is said to be finite (respectively σ-finite) if ν() < (respectively there exists a sequence ( i ) i M such that = i=1 i and ν( i ) < for each i). (v) A real valued signed measure ν : M R, that is, if ν(e) R for each E M, is called a real measure. Example 1.35 (Examples of signed measures). Let us introduce below two remarkable examples of signed measures on a given measure space (, M). i=1 (i) Let µ : M [0, ] be a measure and let f : R be a measurable function. Suppose at least one of f + := f 0 or f := ( f) 0 is integrable, and let ν : M R denote the extended real-valued function on M defined by ν(e) := f dµ E M. E Then is easy to see that ν is a signed measure and ν << µ. If both f + and f are integrable, or, equivalently, f is integrable, then ν is a real measure. (ii) Let µ 1, µ 2 : M [0, ] be measures and assume that al least one of them is finite. Let ν : M R denote the extended real-valued function on M defined by ν(e) := µ 1 (E) µ 2 (E) E M. Then is easy to see that ν is a signed measure. If both µ 1 and µ 2 are finite, then ν is a real measure. Remark 1.36. Observe that a measure is a signed measure. In some contexts we will emphasize that a measure µ is not a signed measure by saying that it is a positive measure. Notice also that a signed (or also real) vector measure ν is not an increasing set function. 13

14 Exercise: ν() <. A signed measure ν is a real measure if and only if it is finite, that is Theorem 1.37 (Lebesgue decomposition theorem for signed measures). Let (, M, µ) be a measure space with µ σ-finite, and ν : M R be a σ-finite signed measure. Then there are two signed measures ν ac, ν s : M R such that (LD) ν ac << µ, ν s µ, ν = ν ac + ν s, and there exists a measurable function w : R such that either w + or w is integrable with respect to µ such that (RN) ν ac (E) = w dµ E M. Moreover both decomposition (LD) and representation (RN) are unique. Proof. See [F, Theorem 3.8]. E Remark 1.38. Notice that the sum of signed measures ν ac + ν s is well defined in (LD) since ν ac and ν s are mutually singular. Remark 1.39. Suppose ν is a real measure (observe that ν is also σ-finite), that is ν : M R, and ν << µ with µ a given σ-finite positive measure on M. Applying Theorem 1.37, we have that ν(e) = ν ac (E) = w dµ E M, and w : R is now an integrable function on with respect to µ, that is w dµ <. Indeed, since ν(e) R for each E M, it is easy to see that w must be integrable. We recommend [GZ, Section 6.5] and [F, Chap. 6] for a complete treatment concerning signed measures. However we point out that signed measures in Example 1.35 are really the only examples: every signed measure can be represented in either of these two forms. An other important tool in GMT will turn out to be the notion of signed vector measure, which is an extension of the one of signed measure. Definition 1.40 (Vector signed measures). Let (, M) be a measure space. (i) A vector set function ν = (ν (1),..., ν (m) ) : M R m is a vector signed measure if its components ν (i) : M R (i = 1,..., m) are signed measures (according to Definition 1.34). (ii) A vector signed measure ν : M R m is a vector measure if it is R m -valued vector measure, that is ν : M R m. (iii) If ν is a signed vector measure, we define its total variation ν : M [0, ] as follows: { } ν (E) = sup ν(e h ) : (E h ) h M pairwise disjoint, E = h=1e h, where h=1 v := E { v R m if v R m if v R m \ R m.

(iv) If ν is a real measure, that is ν : M R, we define its positive and negative parts respectively as follows: ν + = ν + ν and ν = ν ν 2 2 Notation: In the following, we will say countable partition of a set E a pairwise disjoint sequence of sets (E h ) h such that h=1 E h = E. Remark 1.41. Observe that, according to Definition 1.34 (i), a R m -valued signed vector measure ν = (ν (1),..., ν (m) ) satisfies the following two properties: (SVM1) ν( ) = 0 := (0,..., 0) R m ; (SVM2) For each sequence of disjoint sets (E h ) h M, it holds that ( ) ν( h=1e h ) = ν(e h ) := ν (1) (E h ),..., ν (m) (E h ), h=1 h=1 where the series on the right-hand side h=1 ν(i) (E h ) (i = 1,..., m) either converges absolutely or diverges to or +. Notice also that, if ν is R m -valued vector measure, then the absolute convergence in the series in (SVM2) is a requirement on the set function ν: in fact the sum of the series cannot depend on the order of its terms, as the union does not. Observe also that, when m = 1, the notion of signed vector measure (respectively vector measure) agrees with the one of signed measure (respectively real measure). Eventually notice that a R m -valued set function ν = (ν 1,..., ν m ) : M R m is a vector measure if and only if ν i : M R (i = 1,..., m) is a real measure. Remark 1.42. The introduction of the notion of total variation solves the problem of finding a positive measure µ which dominates a given signed vector measure ν on M in the sense that ν(e) µ(e) for each E M, looking for keeping µ as small as we can. Every solution to this problem (if there is one at all) must satisfy µ(e) = µ(e h ) ν(e h ) h=1 for each partition (E h ) h of any set E M, so that µ(e) is at least equal to quantity ν (E). This suggest the reason of total variation s definition like in Definition 1.40 (iii). Let us show that the total variation of a signed vector measure (respectively vector measure) is a positive measure (respectively positive finite measure). Theorem 1.43. (i) Let ν be a signed vector measure on (, M). Then its total variation ν is a positive measure. (ii) If ν is a vector measure, then ν is a positive finite measure, that is ν () <. Proof. (i) We have to prove that (1.10) ν ( ) = 0 and h=1 (1.11) ν is countable additive. h=1 15

16 It is trivial, by definition, that (1.10) holds. Let us show (1.11). Let us firstly observe that ν : M [0, ] is increasing, that is (1.12) ν (E) ν (F ) if E F. Indeed let (E h ) h M be a partition of E, then the family of sets {E h : h} {F \ E} is a countable partition of F. Thus ν(e h ) ν(e h ) + ν(f \ E) ν (F ). h=1 h=1 Then, taking the supremum on the partitions of E in the previous inequality, we get (1.12). Let us now show that (1.13) ν is countably subadditive and (1.14) ν is additive. Observe that, from (1.12), (1.13) and (1.14), (1.11) follows. Indeed let (E h ) h M be pairwise disjoint. Then by countable subadditivity (1.13), (1.15) ν ( h=1e h ) ν (E h ). On the other hand, by (1.12) and (1.14), (1.16) ν ( h=1e h ) ν ( m h=1e h ) = h=1 m ν (E h ) m N. Thus, by (1.15) and (1.17), (1.11) follows. Let us now prove (1.13). Let E, (E h ) h be in M such that E h=1 E h. Let us define a pairwise disjoint sequence (E h ) h M in the following way E 0 := E 0 and E h := E h\ h 1 i=1 E i if h 1. Let (F i ) i be a partition of E, since (E h F i) h is a partition of F i for fixed i, by countable additivity (SVM2) and (1.12), we can infer ν(f i ) = ν(e h F i ) ν(e h F i ) i=1 i=1 h=1 i=1 h=1 = ν(e h F i ) ν (E h E) ν (E h ). h=1 i=1 Taking the supremum on the partitions of E in the previous inequality, it follows that ν (E) ν (E h ) h=1 and (1.13) follows. Let us now prove (1.14). Let E, F M be disjoint. If at least one between ν (E) and ν (F ) is, then, by (1.12), it is immediate that h=1 h=1 ν (E F ) = = ν (E) + ν (F ). h=1

Thus, without loss of generality, we can assume that both ν (E) and ν (F ) are finite. By definition, for each ɛ > 0, there exist a partition (E h ) h M of E and one (F h ) h M of F such that ν (E) ν (E h ) + ɛ, h=1 ν (F ) ν (F h ) + ɛ. Observe now that the family of sets {E h : h N} {F h : h N} is a countable partition of E F. Then, by the previous inequality, it follows that, for each ɛ > 0, ν (E) + ν (F ) 2ɛ ν (E h ) + ν (F h ) ν (E F ). h=1 By countable subadditivity (1.13) and the previous inequality, (1.14) follows. (ii) It is sufficient to assume that ν is a real measure, that is m = 1 and ν : M R. The R m -valued case being an easy consequence of the following estimate m ν (E) ν i (E) E M, i=1 if ν = (ν 1,..., ν m ). Suppose that for some E M has ν (E) =. Let us then prove there exist two disjoint sets A, B M such that (1.17) E = A B, ν(a) > 1, ν(b) > 1, either ν (A) = or ν (B) =. h=1 h=1 By definition, there is a partition (E h ) h of E such that m (1.18) ν(e h ) > 2( ν(e) + 1). h=1 Let I := {1 h m : ν(e h ) > 0} and J := {1 h m : ν(e h ) < 0}. Since, by the additivity m ν(e h ) = ν(e h ) ν(e h ) = ν( h I E h ) ν( h J E h ), h=1 h I h J by (1.18), we can infer that either ν( h I E h ) = ν( h I E h ) > ( ν(e) + 1) or ν( h J E h ) = ν( h J E h ) > ( ν(e) + 1). Let A denote one between sets h I E h and h J E h such that ν(a) > ( ν(e) + 1) and let B := E \ A. Then ν(b) = ν(e) ν(a) ν(a) ν(e) > 1. By the additivity of ν, it is clear that either ν (A) = or ν (B) =. Therefore (1.17) follows. Now if ν () =, then we can apply (1.17) with E = and split into two sets A 1 and B 1 with ν(a 1 ) > 1 and ν (B 1 ) =. Split B 1 into two sets A 2 and B 2 with ν(a 2 ) > 1 and ν (B 2 ) =. Continuing in this way, we get a countably infinite disjoint family of sets (A h ) h with ν(a h ) > 1 for each h. The countable additivity of ν implies that ν( h=1a h ) = ν(a h ). But this series cannot converge since ν(a h ) does not tend to 0 as h. contradiction shows that ν () <. h=1 17 This

18 Remark 1.44. The above theorem shows that for any real measure ν, its positive and negative part are positive finite measures, hence the decomposition ν = ν + ν holds; it is known as the Jordan decomposition of ν. We point out that a Jordan decomposition still hold for signed measures, by means of a suitable notion of positive and negative parts for a signed measure (see [GZ, Theorem 6.31] or [F, Theorem 3.4]). Corollary 1.45. Let ν : M R be a signed measure. Then ν is σ- finite if and only if so does its total variation ν : M [0, ]. Proof. If ν is σ-finite, since ν(e) ν (E) E M, according to Definition 1.34 (iv), ν is also σ-finite. Suppose that ν is σ- finite, that is there is a disjoint sequence ( k ) k M such that ν( k ) < for each k. For given k, let us define the set function ν k : M R defined by ν k (E) := ν(e k ). Notice that ν k is a real measure. Thus, from Theorem 1.43 (ii), its total variation ν k is a positive finite measure, that is ν k () <. Let us now prove that (1.19) ν k () = ν ( k ) from which it will follow that ν is σ-finite and the proof is accomplished. By definition { } ν k () = sup ν(e h k ) : (E h ) h M partition of (1.20) h=1 { } sup ν(f h : (F h ) h M partition of k = ν ( k ). h=1 Let (F h ) h M be a partition of k and define the partition of as E 1 := \ k, E h := F h 1 if h 2.Then it trivial that (1.21) ν(f h ) = ν(e h k ) = ν k (E h ) ν k (). h=1 h=1 Therefore, by (1.20) and (1.21), (1.19) follows. Remark 1.46. It is immediate to check that R m -valued vector measures can be added and multiplied by real numbers, hence they form a real vector space; moreover, an easy consequence of Theorem 1.43 is that the total variation is a norm on the space of measures, which turns out to be a Banach space. If is a locally compact separable metric space, it will be identified with the dual of a space of continuous functions and this will give the completeness in another way (see Corollary 1.82 and Theorem 1.87). Example 1.47. According to the notation for positive measure (see (RN)), given a measure space (, M, µ) and a vector function w = (w 1,..., w m ) : R m, with each w i : R (i = 1,..., m) measurable functions such that either w i,+ or w i, is integrable. Let us define the vector set function µ w : M R m defined as follows ( ) (1.22) µ w (E) = w dµ := w 1 dµ,..., w m dµ E M. E E h=1 E

Then it is easy to see that µ w is a signed vector measure and its total variation is computed in the following proposition. Proposition 1.48. Let (, M, µ) be a measure space and let w = (w 1,..., w m ) : R m, with each w i : R (i = 1,..., m) measurable functions such that either w i,+ or w i, is integrable. Consider the vector signed measure µ w in (1.22). Then (1.23) µ w (E) = w dµ E M. E Proof. It is easy to see that, by definition of total variation for a vector measure, µ w (E) w dµ E M. E Let us prove the reverse inequality. Let E M. If µ w (E) = we are done. Then suppose µ w (E) <. Note that, from this assumption, we have that each w i (i = 1,..., m) is integrable on E, that is E w i dµ <. Without loss of generality, we can assume that each w i is a real-valued function on E and then we can consider w = (w 1,..., w m ) : E R m. Let D = {z h : h N} be a dense set in the unit sphere S m 1 := {y R m : y = 1}. For any ɛ (0, 1) let us define σ : E N σ(x) := min {h N : w(x), z h (1 ɛ) w(x) } x E, and let E h := σ 1 (h). Then (E h ) h M is a pairwise disjoint sequence and E = h=0 E h. Therefore (1 ɛ) w dµ = (1 ɛ) w dµ E h=0 E h w(x), z h dµ w(x) dµ µ w (E) ɛ (0, 1). E h E h h=0 h=0 Thus, getting ɛ 0 in the previous inequality, the proof is accomplished. Definition 1.49 (Integrals). Let (, M) be a measure space. (i) Let ν : M R be a real measure. If u : R is a ν -measurable function, we say that u is ν-integrable if u is ν -integrable and we set u dν := u dν + u dν. If u = (u 1,..., u k ) : R k is a ν -measurable vector function,we say that u is ν-integrable if each its component u i (i = 1,..., k) is ν -integrable and we set ( ) u dν := u 1 dν,..., u k dν. 19

20 (ii) Let ν = (ν 1,..., ν m ) : M R m be a vector measure. If u : R is a ν -measurable function, we say that u is ν-integrable if u is ν -integrable and we set ( ) u dν := u dν 1,..., u dν m. (iii) Let E M, the integral of of a function u on E is defined by u dν := u χ E dν, provided that the right-hand side makes sense. E Remark 1.50. Notice that an immediate consequence of the above definition is the inequality u dν u d ν which holds for every extended real or vector valued summable function u and for every positive, real or vector measure ν. Definition 1.51 (Absolute continuity and singularity for signed vector measures). Let (, M) be a measure space. (i) Let µ be a positive measure and ν be a vector signed measure on the measure space (, M). We say that ν is absolutely continuous with respect to µ, and write ν << µ, if ν << µ, as positive measures, that is, for every E M the following implication holds: µ(e) = 0 ν (E) = 0. (ii) If µ or ν are signed R m -valued signed measures on measure space (, M), we say that they are mutually singular, and write µ ν, if µ and ν are mutually singular, as positive measures, that is, there exists E M such that µ (E) = ν ( \ E) = 0. Remark 1.52. Observe that, given a positive measure µ on a measure space (, M), then vector measure µ w defined in (1.22) is trivially absolutely continuous with respect to µ by Proposition 1.48. Lebesgue decomposition theorem for vector signed measures 1.53. Let ν and µ be respectively a R m -valued σ-finite measure and a σ-finite positive measure on a measure space (, M). Then there is a decomposition of ν such that (1.24) ν = ν ac + ν s, where ν ac and ν s are still R m -valued signed measures on (, M) with ν ac << µ and ν s µ. The decomposition is unique. Moreover there exists a unique vector function w = (w 1,..., w m ) : R m with either w i,+ or w i, (i = 1,..., m) integrable functions w.r.t. µ such that (1.25) ν ac (E) = µ w (E) = w dµ E M. E

Proof. Let ν = (ν 1,..., ν m ) : M R m. By definition, for each i = 1,..., m, ν i : M R is a σ-finite signed vector measure. By Theorem 1.37, there are two signed measures ν i,ac, ν i,s : M R such that (1.26) ν i,ac << µ, ν i,s µ, ν i = ν i,ac + ν i,s, and there exists a measurable function w i : R such that either w i,+ or w i, is integrable with respect to µ such that (1.27) ν i,ac (E) = w i dµ E M. Moreover both decomposition (1.26) and representation (1.27) are unique. Define the signed vector measures ν ac, ν s : M R m E ν ac := (ν 1,ac,..., ν m,ac ) and ν s := (ν 1,s,..., ν m,s ) and the vector function w := (w 1,..., w m ) : R m. Then it is trivial to see that (1.24) and (1.25) now hold for the signed vector measure ν. Therefore the proof is accomplished. Each signed vector measure ν is trivially absolutely continuous with respect to its total variation ν, the following useful decomposition for vector measures immediately follows from the Lebesgue decomposition theorem for signed vector measures 1.53, Proposition 1.48 and Remark 1.39. Corollary 1.54 (Polar decomposition for vector measures). Let ν be a R m -valued measure on the measure space (, M). Then there exists a unique measurable vector function w ν : R m with w ν (x) = 1 ν a.e. x such that ν = ν wν, that is ν(e) = w ν d ν E M.. E Proof. Let us first notice that, by Theorem 1.43 (ii), both ν and ν are finite and ν << ν. By Theorem 1.53 and Remark 1.39 there exists a vector function w ν : R m such that (1.28) ν(e) = w ν d ν E M. By (1.28) and Proposition 1.48, we can infer that ν (E) = w ν d ν E M. E E Since ν is finite, we have that w ν (x) = 1 ν -a.e. x and the proof is accomplished. 21

22 1.5. Spaces L p (, µ) and their main properties. Riesz representation theorem. Completeness and dual space of L p (, µ) In this subsection we will only request that (, M, µ) is a measure space. Let us introducel the space of p-integrable functions with respect to measure µ. Definition 1.55. Let p [1, ], where L p (, µ) := { f : R : f is measurable and f L p < + } f L p = f L p (,µ) := ( f(x) p dµ(x)) 1/p if 1 p < inf {M > 0 : f(x) M µ a.e. x } if p = The quantity f L p is called the L p norm of f on measure space (, M, µ). When = Ω is an open subset of R n, µ = L n, M = Ω M n, where M n denotes the class of n-dimensional Lebesgue measurable sets of R n and d the Euclidean distance, we will simply denote L p (, µ) as L p (Ω). Remark 1.56. When dealing with measure- theoretic or functional-analytic propenies of functions and L p spaces, it is often convenient to consider functions that agree a.e. as identical, thinking of the elements of L p spaces as equivalence classes; in particular, this makes L pa norm. We shall follow this path whenever our statements will depend only on the equivalence class without further mention, provided that this is clear from the context. Let us recall the following fundamental result concerning the completeness of L p (see [GZ, Theorem 6.24]). Theorem 1.57 (Fisher-Riesz,1907). (L p (, µ), L p) is a B.s. if 1 p. Moreover L 2 (, µ) turns out to be a Hilbert space with respect to the scalar product (f, g) L 2 := f g dµ f, g L 2 (, µ). As a consequence of the proof of Riesz- Fisher s Theorem we have the following useful result. Theorem 1.58. Let (f h ) h L p (, µ) and f L p (, µ) with 1 p. Suppose that (MC) lim f h f L p (,µ) = 0. h Then, there exist a subsequence (f hk ) k and a function g L p (, µ) such that (i) f hk (x) f(x) µ a.e. x ; (ii) f hk (x) g(x) µ a.e. x, k. Proof. See [GZ, Theorem 6.25]. Remark 1.59. It does not hold the implication (MC) f h (x) f(x) µ-a.e. x Ω..

Historical notes: ([P, Sections 1.1.4,1.5.2,4.4.1]) Fisher and Riesz invented the Hilbert space L 2 ([a, b]) in 1907, by proving its completeness. Both authors observed the significance of Lebesgue s integral as the basic ingredient. Subsequently, in 1909, Riesz extended this definition to exponents 1 < p < and described how the interval [a, b] can be replaced by any measurable set of R n. L p spaces are sometimes called Lebesgue spaces, named after H. Lebesgue (Dunford & Schwartz 1958, III.3), although according to Bourbaki (1987) they were first introduced by Riesz. Definition 1.60. Let 1 p < and denote { p p if 1 < p < := p 1 if p = 1 p is called conjugate exponent of p. Theorem 1.61 (Hölder inequality). Let p and p be conjugate exponents, 1 p < Let f L p (, µ) and g L p (, µ). Then f g L 1 (, µ) and fg L 1 (,µ) f L p (,µ) g L p (,µ) Proof. See [R1, Theorem 3.8] or [GZ, Theorem 6.20]. The Hölder inequality establishes a duality between L p (, µ) and the dual space of L p (, µ), denoted (L p (, µ)) according to the notation of functional analysis: if u L p (, µ), it is well defined the continuos linear functional T (u) : L p (, µ) R, that is T (u) (L p (, µ)), by T (u)(f) := T (u), f (L p (,µ)) L p (,µ) := u f dµ f L p (, µ). The question naturally arises whether all continuous linear functionals on L p (, µ) have this form, and whether the representation is unique. The answer is affirmative if 1 < p <. It is also affirmative if p = 1, provided that an additional condition on measure µ. Riesz representation theorem for the dual space of L p 1.62. If 1 < p <, then the mapping T : L p (, µ) (L p (, µ)), defined by T (u), f (L p (,µ)) L p (,µ) := u f dµ f L p (, µ), is an isometric isomorphism, that is, T is a linear, one-to-one, onto mapping and T (u) (L p (,µ)) = u L p (,µ) u L p (, µ). If p = 1, the same conclusion holds under the additional assumption that µ is σ-finite. We will mean this feature by means of the identification (1.29) L p (, µ) (L p (, µ)). Proof. See [GZ, Theorem 6.43] or [F, Theorem 6.15], and [R1, Theorem 6.16]) if p = 1 provided µ is σ-finite. Remark 1.63. Identification (1.29) may fail in the other cases: see [F, section 6.2]. 23

24 Historical notes:([p, Section 2.2.7]) The identity (L p (a, b)) = L p (a, b) with 1 < p < was proved by F. Riesz in 1909. The limit case p = 1 is due to Steinhuas in 1919. Density of continuous functions in (L p (, µ), L p). Riesz representation theorem. The subject of this subsection concerns measure and integration theory on locally compact metric spaces. We have seen that the Lebesgue measure on R n interacts nicely with the topology on R n - measurable sets can be approximated by open or compact sets, and integrable functions can be approximated by continuous functions - and it is of interest to study measures having similar properties on more general spaces. Moreover, it turns out that certain linear functionals on spaces of continuous functions are given by integration against such measures. This fact constitutes an important link between measure theory and functional analysis, and it also provides a powerful tool for constructing measures. Let us begin to deal with the approximation of continuous functions in L p (, µ) when (, d) is a general separable metric space equipped by a locally finite Borel measure µ. Theorem 1.64 (Approximation in L p by continuous functions). Let (, B(), µ) be a measure space with (, d) separable and µ locally finite. Then C 0 () L p () is dense in (L p (), L p), provided that 1 p <. Before the proof of Theorem 1.64, we need the so-called Urysohn s lemma in metric spaces. Lemma 1.65 (Urysohn s lemma). Let (, d) be a metric space, let F and U be, respectively, a closed set and an open set such that F U. Then there exists a function ϕ C 0 () such that (1.30) 0 ϕ 1, ϕ 1 in F and ϕ 0 in \ U. Proof. See [F, Exercise 3 in Sect. 4.1 and Lemma 4.15]. Lemma 1.66 (Approximation in L p of characteristic functions by continuous functions ). Let (, d) be a metric space, let µ be a Borel measure on (, d) and let V be an open set in (, d) such that µ(v ) <. Then for each B B() with B V and δ > 0, there exists a function ϕ C 0 () such that 0 ϕ χ V, and χ B ϕ L p (,µ) < δ. Proof. By the approximation of Borel sets by means of closed and open sets (see Theorem 1.9 and Remark 1.11), for each δ > 0, there exist a closed set F and on open set U V such that (1.31) F B U and µ (U \ F ) < δ. By Urysohn s lemma 1.65, there exists a function ϕ C 0 () satisfying (1.30).Since by (1.31), we get the desired conclusion. χ B ϕ χ U χ F,

Proof of Theorem 1.64. We have to prove that, for each ɛ > 0 and f L p (, µ), there exists ϕ C 0 () such that (1.32) f ϕ L p (,µ) < ɛ. By Theorem 1.18 (ii), the exists an increasing sequence of open sets (V i ) i such that (1.33) = i=1v i, and µ(v i ) < for each i. By (1.33) and Beppo Levi s theorem, there exists an integer ī = ī(ɛ) such that ( ɛ ) p (1.34) f p L p (\V,µ) = ī f p dµ <. 3 \V ī Let us now consider f as a function f : V ī R. By the density of simple functions in L p (V ī, µ) (see Theorem 1.21 and [GZ, Theorem 5.24]), there exists a simple function m s(x) s ɛ (x) = c j χ Aj (x) x V ī, such that j=1 (1.35) f s L p (V ī,µ) < ɛ 3, where A j B(V ī ) B() and A j V ī, c j R (j = 1,..., m). Let M := max { c j : j = 1,..., m}. By Lemma 1.66, for each j = 1,..., m, there exists a function ϕ j C 0 () such that (1.36) ϕ j 0 in \ V ī and χ Aj ϕ j L p (V ī,µ) < ɛ 3Mm. By (1.36), it follows that, if ϕ := m j=1 c j ϕ j, (1.37) s ϕ L p (,µ) = s ϕ L p (V ī,µ) < ɛ 3. 25 Thus, by (1.34), (1.35) and (1.37), we get (1.32). From now on we will request that (, B(), µ) is a measure space with (, d) locally compact, separable metric space (we will often use the abbreviation l.c.s. in the following) and µ a Radon measure. Note that, under the above assumptions, C 0 c() L p (, µ) for each p [1, ], provided that µ is a Radon measure on. On a locally compact metric space, Theorem 1.64 can be strenghten as follows. Theorem 1.67 (Approximation in L p by continuous functions in locally compact spaces ). Let (, B(), µ) be a measure space with (, d) l.c.s. and µ Radon measure. Then C 0 c () is dense in (L p (), L p), provided that 1 p <. Proof of Theorem 1.67. The proof can be carried out as in the case of L p (Ω), by means of the approximation by simple functions ( Theorem 1.21) and Lusin s theorem (Theorem 1.24): see [R1, Theorem 3.14] and also [SC, Theorem 2.59].