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14. Lebesgue Integral Dered Numbers Cn, Cn < Cn+1, Which Is Finite Or 106 2. THE LEBESGUE INTEGRATION THEORY 14. Lebesgue integral 14.1. Piecewise continuous functions on R. Consider a collection of or- dered numbers cn, cn < cn+1, which is finite or countable. Suppose that any interval (a,b) contains only finitely many numbers cn. Con- sider open intervals Ωn =(cn,cn+1). If the set of numbers cn contains the smallest number m, then the interval Ω− = (−∞,m) is added to the set of intervals {Ωn}. If the set of numbers cn contains the greatest + number M, then the interval Ω = (M, ∞) is added to {Ωn}. The − union of the closures Ωn =[cn,cn+1] (and possibly Ω =(−∞,m] and Ω+ =[M, ∞)) is the whole real line R. So, the characteristic properties of the collection {Ωn} are: 0 (i) Ωn ∩ Ωn0 = ∅ , n =6 n (ii) (a,b) ∩{Ωn} = {cj,cj+1,...,cm} , (iii) Ωn = R , [n Property (ii) also means that any bounded interval is covered by finitely many intervals Ωn. Alternatively, the sequence of endpoints cn is not allowed to have a limit point. For example, let cn = n where n is an integer. Then any interval (a,b) contains only finitely many integers. The real line R is the union of intervals [n, n+1] because every real x either lies between two integers or coincides with an integer. If cn is the collection of all non-negative integers, then R is the union of I− = (−∞, 0] and all [n, n + 1], n = 1 0, 1,.... However, the collection cn = n , n = 1, 2,..., does not have the property that any interval (a,b) contains only finitely many elements cn because 0 <cn < 2 for all n. Continuous extension. A collection of all functions that are continuous on an open interval (a,b) is denoted by C0(a,b). The function f ∈ C0(a,b) is said to have a continuous extension to the boundary points a and b if the limits lim f(x)= f+(a) , lim f(x)= f−(b) x→a+ x→b− exist. In this case, f ∈ C0([a,b]). Definition 14.1. (A piecewise continuous function) A function f : R → R is said to be piecewise continuous on R if there exists an at most countable collection of open intervals Ωn with no common points such that any bounded interval is covered by finitely 0 many closed intervals Ωn, and f ∈ C (Ωn). 14.LEBESGUEINTEGRAL 107 So, a piecewise continuous function is continuous everywhere except possibly at the end points of the intervals x = cn. The function is also required to have the left and right limits at every cn. Thus, a piecewise continuous function has only jump discontinuities, and any bounded interval can contain only finitely many point where the function is not continuous. In particular, one can set f to be constant on each interval from {Ωn}. In this case, f is called piecewise constant. One should also note that a piecewise continuous function is allowed to have values f(cn). They are generally different from the limit values f±(cn). A piecewise continuous function is not continuous on a set measure zero, or it is continuous almost everywhere. Indeed, the set of points cn is a countable set of points, that is, it is the countable union of sets of measure zero and such a set has measure zero. One can also say that a piecewise continuous function is continuous almost everywhere. Not that the converse is false. Therefore any piecewise continuous function is Riemann integrable on any [a,b]. The value of the Riemann integral does not depend on the values of a piecewise continuous function at the points where it is not continuous. 14.2. Measurable functions on R. It was shown before that the limit function of a pointwise convergent sequence of functions does not gen- erally inherits properties of the terms of the sequence. For example, the limit function is not necessarily continuous even if the terms are continuous functions, or the limit function may not be Riemann inte- grable even if the terms are. More generally, let A be a set of functions that is defined by some characteristic property (e.g., continuity, or in- tegrability, etc.). Then the pointwise limit of a convergent sequence {fn}⊂A does not in general belong to A. One can ask how large the set A should be in order to be complete, meaning that the limit of every pointwise convergent sequence in A belongs to A. It turns out that such a set of functions exists and is known as a set of measurable functions. Suppose that a sequence {fn} of functions on R converges pointwise almost everywhere (that is, a numerical sequence {fn(x)} can have no limit for particular x that form a set of measure zero). In this case, one writes lim fn(x)= f(x) a.e. n→∞ For example, lim [cos(πx)]n = 0 a.e. n→∞ 108 2. THE LEBESGUE INTEGRATION THEORY Note that the limit does not exist if x is an integer. If x is not an integer, then | cos(πx)| < 1 and the limit is equal to zero. But the integers form a set of measure zero. So, the limit function is equal to the zero function almost everywhere. Definition 14.2. (A measurable function) A function f is called measurable if it coincides almost everywhere with the limit of an almost everywhere convergent sequence of piecewise continuous functions. Properties of measurable functions. Evidently, every piecewise continu- ous function f is measurable because one can take a sequence of piece- wise continuous functions fn(x) = f(x) of identical terms which ob- viously converges to f(x). Suppose that f is a measurable function and g coincides with f almost everywhere. Then g is also measurable. Indeed, Let fn be a sequence of piecewise continuous functions that converges to f almost everywhere. Since f and g differ only on a set of measure zero, fn converges to g almost everywhere, too: f(x) is measurable ⇒ g(x) is measurable f(x)= g(x) a.e. More generally, one can prove that • a function that is not continuous on a set of measure zero is measur- able. Therefore every Riemann integrable function is measurable by Theo- rem 10.4. Furthermore, every function for which the improper Riemann integral exists is also measurable. So, the set of measurable functions contains all Riemann integrable functions (either in the proper or im- proper sense). There are measurable functions that are not Riemann integrable. For example, the Dirichlet function 1 , x ∈ Q fD(x)= 0 , x∈ / Q is not Riemann integrable on any interval. However, the set Q of rational numbers has measure zero in R. Therefore fD(x) = 0 a.e., but any constant function and, in particular, g(x) = 0 is measurable and, hence, so is the Dirichlet function. Using the basic limit laws, it is not difficult to show that f(x)+ g(x) is measurable f(x) is measurable ⇒ f(x)g(x) is measurable g(x) is measurable f(x)/g(x), g(x) =6 0, is measurable 14.LEBESGUEINTEGRAL 109 Note that if fn(x) and gn(x) are sequences of piecewise continuous func- tions, then the functions fn(x)+ gn(x), fn(x)gn(x), and fn(x)/gn(x), gn(x) =6 0, are also sequences of piecewise continuous functions, and the above assertion follows from the basic laws of limits. A set of mea- surable function is complete relative to addition and multiplication by a number. In other words, a linear combination of measurable functions is measurable. Sets with this property are called a linear space. Thus, the set of measurable functions is a linear space. Given two functions f and g, define the following functions f(x) , f(x) > g(x) max(f, g)(x) = g(x) , f(x) ≤ g(x) g(x) , f(x) > g(x) min(f, g)(x) = f(x) , f(x) ≤ g(x) One can prove that the functions max(f, g) and min(f, g) are measur- able, if f and g are measurable. It follows that the absolute value |f(x)| = max(f, 0)(x) − min(f, 0)(x) of a measurable function f is measurable. Theorem 14.1. (Completeness of the set of measurable functions) A function coinciding almost everywhere with the limit of an almost everywhere convergent sequence of measurable functions is measurable. Thus, the set of measurable functions (and sets) is quite large. It seems that every imaginable function is measurable. So, the question of interest: Are there non-measurable functions and sets? It appears that one can prove that they exist (using the axiom of choice), but no explicit example has been constructed so far! This suggests that all functions and sets that can possibly be used in applications or otherwise are measurable. For this reason, in what follows all sets are assumed to be measurable and all functions are assumed to be measurable and bounded almost everywhere. 14.3. Definition of the Lebesgue integral. To avoid confusion between Riemann and Lebesgue integrals, the Riemann integral will be denoted as b R f(x) dx or R f(x) dx Za Z where the latter is the improper Riemann integral (or absolutely con- vergent Riemann integral) over the whole real line. 110 2. THE LEBESGUE INTEGRATION THEORY Definition 14.3. (The space L+) Let a real function f(x) be the limit of a non-decreasing sequence of piecewise continuous functions fn(x) such that the sequence of Riemann integrals is bounded: fn(x) ≤ fn+1(x) , n = 1, 2,..., , ∀x ∈ R , R fn(x) dx ≤ M , n = 1, 2,..., Z for some number M.
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