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AN INTRODUCTION to GEOMETRIC MEASURE THEORY and an APPLICATION to MINIMAL SURFACES ( DRAFT DOCUMENT) Academic Year 2018/19 Francesco Serra Cassano

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AN INTRODUCTION to GEOMETRIC MEASURE THEORY and an APPLICATION to MINIMAL SURFACES ( DRAFT DOCUMENT) Academic Year 2018/19 Francesco Serra Cassano AN INTRODUCTION TO GEOMETRIC MEASURE THEORY AND AN APPLICATION TO MINIMAL SURFACES ( DRAFT DOCUMENT) Academic Year 2018/19 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 decompo- sition for vector measures. I.5 Spaces Lp(X; µ) and their main properties. Riesz representation theorem. I.6 Operations on measures. I.7 Weak*-convergence of measures. Regularization of Radon measures on Rn. II. Differentiation of Radon measures. II.1 Covering theorems and Vitali-type covering property for measures on Rn. II.2 Derivatives of Radon measures on Rn. Lebesgue-Besicovitch differentiation theorem for Radon measures on Rn. II.3 Extensions to metric spaces. III. An introduction to Hausdorff measures, area and coarea formulas. III.1 Carath´eodory'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; H1 and the classical notion of length in Rn; isodiametric inequality and identity Hn = Ln on Rn; k-dimensional densities. III.4 Area and coarea formulas in Rn and some applications. III.5 Extensions to metric spaces. IV. Rectifiable sets and blow-ups of Radon measures. IV.1 Rectifiable sets of Rn and their decomposition in Lipschitz images. IV.2 Approximate tangent planes to rectifiable sets. IV.3 Blow-ups of Radon measures on Rn 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 Rn, 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 n B) [ (B n A) : We shall tipically work in a metric space X 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 Rn. Here are the basic notations used in metric spaces throughout these notes. The closed and open balls with centre x 2 X and radius r, 0 < r < 1, are denoted by B(x; r) = fy 2 X : d(x; y) 6 rg ; U(x; r) = fy 2 X : d(x; y) < rg : In Rn we also set 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 B^. The diameter of a nonempty subset A ⊂ X is d(A) = diam(A) = sup fd(x; y): x; y 2 Ag : We agree d(;) = 0. If x 2 X and A and B are non-empty subsets of X, the distance from x to A and the distance between A and B are, respectively, d(x; A) = inf fd(x; y): y 2 Ag ; d(A; B) = inf fd(x; y): x 2 A; y 2 Bg : For > 0 the open -neighbourhood of A is I(A) = fx 2 X : d(x; A) < g : If A ⊂ Rn, then jAj = Ln(A) where Ln 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 re- sults of measure theory such as: outer measure, measure, signed measures and vector measure with their properties and relationships; measurable functions and their prop- erties; Lp spaces and Riesz representation theorem; convergence of measures. 3 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 Carath´eodory. 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 X is a set. Definition 1.1. Let X denote a set and P(X) denote the class of all subsets of X. (i) A set function ' : P(X) ! [0; 1] is called an outer measure (o.m.) on X if (OM1) '(;) = 0 ; (OM2) '(A) 6 '(B) if A ⊂ B (monotonicity) ; (OM3) 1 1 X '([i=1Ai) 6 '(Ai) for any sequence (Ai)i ⊂ X (countable subadditivity) : i=1 (ii) A set E ⊂ X is called '-measurable (with respect to an o.m. ' on X), if '(A) = '(A \ E) + '(A n E) 8 A ⊂ X: The class of '-measurable sets will be denoted by M'. (iii)A σ-algebra M on X is a (nonempty) class of subsets M ⊂ P(X) satisfying the two following properties: (σA1) if E 2 M; then X n E 2 M ; 1 (σA2) for each sequence (Ei)i ⊂ M; then [i=1 Ei 2 M : (iv) A measure µ on X is a set function µ : M! [0; 1], where M is a σ-algebra on X, satisfying the following two properties: (M1) µ(;) = 0 ; 1 1 X (M2) µ([i=1Ei) = µ(Ei) (countable additivity) i=1 for each disjoint sequence (Ei)i ⊂ M. The structure composed by the triple (X; M; µ), or also the couple (X; M), is called measure space and the sets contained in M are called measurable sets. (v) Let (X; M; µ) be a measure space. The measure µ is said to be finite, if µ(X) < 1; it is said to be σ-finite, if there exists a sequence (Xi)i ⊂ M such 1 that X = [i=1Xi and µ(Xi) < 1 for each i. (v) Let (X; M; µ) be a measure space. A point x 2 M is said to be an atom if the singleton fxg 2 M and µ(fxg) > 0. The set of atoms of µ will be denoted by Sµ and µ is said to be atomic if Sµ 6= ;. If µ is finite or σ-finite the set of atoms Sµ is at most countable. 4 (vii) A function f : X ! R := [−∞; 1] is called measurable with respect to an o.m. ' (respectively with respect to a measure µ : M! [0; 1]) if f −1(U) is '-measurable (respectively f −1(U) 2 M) for each open set U in R. (viii) A simple function s : X ! 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 k X s(x) = ai χAi (x) 8 x 2 X; i=1 k with ai 2 R, Ai ⊂ X (i = 1; : : : ; k), X = [i=1Ai and Ai \ Aj = ; if i 6= 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 (X; 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 : X ! [−∞; 1], it is possible to make a sense to the value integral of f with respect to µ, denoted Z f dµ 2 [−∞; 1] : X R When the integral is finite, that is X f dµ 2 (−∞; 1), f is said to be integrable or also summable. Example 1.2. Let (X; 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; 1] , #(;) := 0, #(E) as the cardinality of E if it is finite, #(E) = 1 otherwise. (iI) (Dirac measures) With each x 2 X we associate the set function δx : M! [0; 1] defined by δx(E) := 1 if x 2 E, δx(E) := 0 otherwise. If (xh)h ⊂ X P1 and (ch)h ⊂ [0; 1) is a sequence such that the series h=1 ch is convergent, P1 we can define the set function h=1 ch δxh : M! [0; 1] 1 ! X X ch δxh (E) := ch : h=1 fh: xh2Eg Measures of this kind are called purely atomic. Now, we enrich the environment X, by adding a topology and we require compat- ibility between topology and measure. Definition 1.3. Let (X; τ) denote a topological space and denote B(X) the σ-algebra of Borel sets of X, i.e. the smallest σ-algebra of X which contains the open and closed sets of X. (i) An o.m. ' on X is called a Borel o.m. if the class of '-measurable sets M' ⊃ B(X).
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