1 Remarks on Lebesgue Integral

Ivan Avramidi: Notes of Hilbert Spaces 11 <strong>Remarks</strong> on Lebesgue IntegralDefinition 1 Characteristic function of a set A ⊂ X is a mapping χ A :X → {0, 1} defined by⎧⎨ 1, if x ∈ Aχ A (x) =(1.1)⎩ 0, if x ∉ ADefinition 2 For a non-zero function f : R n → R, the set, supp f, of allpoints x ∈ R n for which f(x) ≠ 0 is called the support of f, i.e.Clearly, supp χ A = A.supp f = {x ∈ R n |f(x) ≠ 0} . (1.2)Definition 3 Let I be a semi-open interval in R n defined byI = {x ∈ R n | a k ≤ x k < b k , k = 1, . . . , n} (1.3)for some a k < b k . The measure of the set I is defined to beµ(I) = (b 1 − a 1 ) · · · · (b n − a n ) . (1.4)The Lebesgue integral of a characteristic function of the set I is defined by∫χ I dx = µ(I) . (1.5)Definition 4 A finite linear combination of characteristic functions of semiopenintervalsN∑f = α k χ Ik (1.6)is called a step function.k=1Definition 5 The Lebesgue integral of a step function is defined by linearity∫N∑k=1α k χ Ik dx =N∑α k µ(I k ) . (1.7)Definition 6 A function f : R n → R is Lebesgue integrable if ∃ a sequenceof step functions {f k } such thatf ≃k=1∞∑f k , (1.8)k=1

Ivan Avramidi: Notes of Hilbert Spaces 2which means that two conditions are satisfieda)∞∑∫|f k | dx < ∞ (1.9)k=1∞∑b) f(x) = f k (x) ∀x ∈ R n such thatk=1The Lebesgue integral of f is then defined by∫k=1∞∑|f k (x)| < ∞ . (1.10)k=1∞∑∫f dx = f k dx (1.11)Proposition 1 The space, L 1 (R n ), of all Lebesgue integrable functions on R nis a vector space and ∫ is a linear functional on it.Theorem 1a) If f, g ∈ L 1 (R n ) and f ≤ g, then ∫ fdx ≤ ∫ gdx.b) If f ∈ L 1 (R n ), then |f| ∈ L 1 (R n ) and | ∫ f dx| ≤ ∫ |f|dx.Theorem 2 If {f k } is a sequence of integrable functions and∞∑f ≃ f k , (1.12)k=1then∫∞∑∫f =k=1f k , (1.13)Definition 7 The L 1 -norm in L 1 (R n ) is defined by∫||f|| = |f|dx (1.14)Definition 8 A function f is called a null function is it is integrable and||f|| = 0. Two functions f and g are said to be equivalent if f − g is a nullfunction.Definition 9 The equivalence class of f ∈ L 1 (R n ), denoted by [f], is the setof all functions equivalent to f.Remark. Strictly speaking, to make L 1 (R n ) a normed space one has to considerinstead of functions the classes of equivalent functions.Definition 10 A set X ⊂ R n is called a null set (or a set of measure zero)if its characteristic function is a null function.

Ivan Avramidi: Notes of Hilbert Spaces 3Theorem 3a) Every countable set is a null set.b) A countable union of null sets is a null set.c) Every subset of a null set is a null set.Definition 11 Two integrable functions, f, g ∈ L 1 (R n ), are said to be equalalmost everywhere, f = g a.e., if the set of all x ∈ R n for which f(x) ≠ g(x)is a null set.Theorem 4∫f = g a.e ⇐⇒ ||f − g|| =|f − g| = 0 (1.15)Theorem 5 The space L 1 (R n ) is complete.