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106 2. THE LEBESGUE INTEGRATION THEORY

14. Lebesgue 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 (a,b) contains only finitely many numbers cn. Con- sider open intervals Ωn =(cn,cn+1). If the 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 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

Continuous extension. A collection of all functions that are continuous on an open interval (a,b) is denoted by C0(a,b). The 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 ) 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 zero, or it is continuous . Indeed, the set of points cn is a 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 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 ) 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 of measurable functions is measurable. Sets with this property are called a linear . 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 |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 , 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. The limit of the non-decreasing sequence of Rie- mann integrals is called the Lebesgue integral of f and is denoted by the symbol f(x)dx so that R f(x) dx = lim R fn(x) dx . Z n→∞ Z

The set of all such functions is denoted by L+. Note that by the basic law for limits, any linear combination of functions from L+ belongs to L+. So, L+ is a linear space. One can prove that3

• the Lebesgue integral of f ∈ L+ does not depend on the choice of the sequence {fn}.

So, in order to establish whether or not f is Lebesgue integrable, it is sufficient to find at least one non-decreasing sequence of piecewise con- tinuous functions {fn} that converges to f almost everywhere and has a bounded sequence of Riemann integrals. The value of the Lebesgue integral is given by the limit of the sequence of Riemann integrals which always exists. Recall that a non-decreasing bounded numerical sequence always has the limit. Definition 14.4. (Lebesgue integral) A function f is called Lebesgue integrable if it can be represented as the difference of two functions from the set L+:

f(x)= f1(x) − f2(x) , f1 ∈ L+ , f2 ∈ L+ The number

f1(x) dx − f2(x) dx = f(x) dx Z Z Z is called the Lebesgue integral of the function f. The set of all Lebesgue integrable functions is denoted by L.

3see, e.g., V.S. Vladimirov, Equations of Mathematical Physics, Sec. 1.4. 14.LEBESGUEINTEGRAL 111

In order to establish that a given function belongs to L or not, one has to find at least one pair of functions f1 and f2 from L+ whose difference coincides with f. The value of the Lebesgue integral (if it exists) does not depend on the choice of f1 and f2. Indeed, suppose that f1(x) − f2(x)= f(x)= g1(x) − g2(x) , fi ∈ L+ , gi ∈ L+ , i = 1, 2 . It follows from the basic laws for limits, that the Lebesgue integral is additive: f + g ∈ L f ∈ L , g ∈ L ⇒ + + + (f(x)+ g(x)) dx = f(x) dx + g(x) dx R R R By the additivity of the Lebesgue integral for functions from L+ and the relation f1 + g2 = g1 + f2, one infers that

f1(x)dx − f2(x)dx = g1(x)dx − g2(x)dx Z Z Z Z Thus, the Lebesgue integral of f does not depend on the decomposition of f into the difference of two functions from L+. The Lebesgue integral of a complex-valued function f is the f(x)dx = Re f(x) dx + i Im f(x) dx , Z Z Z provided the real and imaginary parts of f are Lebesgue integrable. Definition 14.5. (The Lebesgue integral over a set) A function f is said to be Lebesgue integrable on a measurable set S, f ∈ L(S), if fχS ∈ L, where χS is the characteristic function of S. The number

f(x)χS (x) dx = f(x) dx Z ZS is called the Lebesgue integral of f over S.

Lebesgue integral of a continuous function. Let f be a continuous func- tion on an open interval (a,b) (bounded or not, that is, a = −∞ and b = ∞ are allowed). Then f is Lebesgue integrable on (a,b), f ∈ L(a,b), if and only if, its Riemann integral over (a,b) is absolutely convergent, and in this case, the Lebesgue and improper Riemann in- tegrals coincide: b b f(x) dx = R f(x) dx Za Za 112 2. THE LEBESGUE INTEGRATION THEORY

Note that f is not required to be bounded as x → a+ or x → b−. So, the Riemann integral must be considered in the improper sense. Let us assume first that f(x) ≥ 0. Put

fn(x)= χn(x) f(x) where χn(x) is the characteristic function of the interval Ωn =[an,bn] ⊂ (a,b) where an is monotonically decreasing and lim an = a, while bn is monotonically increasing and lim bn = b. Since f is non-negative, the sequence is monotonically increasing

Ωn ⊂ Ωn+1 ⇒ fn(x) ≤ fn+1(x) 0 Clearly, {Ωn} is an exhaustion of (a,b). Since f ∈ C (Ωn) its Riemann integrals over Ωn exist and form monotonically increasing sequence

bn bn+1 R fndx = R f dx ≤ R f(x) dx = R fn+1dx Z Zan Zan+1 Z The sequence converges if and only if it is bounded:

bn f(x) dx ≤ M, ∀n Zan and the limit is the absolutely convergent Riemann integral of f. Its value is independent of the choice of sequences {an} and {bn}. There- fore f ∈ L+(a,b) and its Lebesgue integral is equal to the (improper) Riemann integral. If f(x) is continuous, then its absolute value |f(x)| is also continu- ous. Define two non-negative continuous functions 1 f±(x)= |f(x)|± f(x) ≥ 0 . 2  Now recall that the Riemann integral of f over (a,b) converges abso- lutely if and only if

b bn R f±(x) dx = lim R f±(x) dx < ∞ n→∞ Za Zan for some choice of the sequences {an} and {bn}. This implies that f± ∈ L+(a,b) and b b b f(x) dx = R f+(x) dx − R f−(x) dx Za Za Za Thus, for continuous functions, the Lebesgue integral coincides with the absolutely convergent Riemann integral. 14.LEBESGUEINTEGRAL 113

14.4. Lebesgue integral in RN . The Lebesgue integral in any Euclidean space is defined in the same way, that is, as a limit of Riemann integrals of piecewise continuous functions.

Piecewise continuous functions on RN . A set is connected if any two points of it can be connected by a piece-wise smooth curve that lies in the set. Definition 14.6. (A region in a Euclidean space) An open and connected set is called a region in RN . Suppose f is continuous on an open set Ω and suppose that the limit of f(x) as x → y ∈ ∂Ω exists for all y from the boundary of Ω. Then f is said to admit a continuous extension to the closure Ω such that f(y) = lim f(x) , x ∈ Ω , y ∈ ∂Ω . x→y In other words, a continuous function is defined for every y ∈ ∂Ω by the value of the limit. In this case, one writes f ∈ C0(Ω). Definition 14.7. Spaces Cp(Ω) and Cp(Ω) Let Ω be a region. Real-valued (or complex-valued) functions f on Ω that are continuous and have continuous partial Dν f, |ν| ≤ p (0 ≤ p < ∞) are said to form the space Cp(Ω). Functions from Cp(Ω) for which f and partial derivatives Dν f, |ν| ≤ p, admit continuous extensions to Ω are said to form the space Cp(Ω). A function f is called piecewise continuous in RN if

(i) there is at most countably many non-intersecting regions Ωn, n = 1, 2,..., (ii) with piecewise smooth boundaries ∂Ωn, (iii) any ball is contained in the union of finitely many closed re- gions Ωn, N (iv) the union of Ωn coincides with R , and 0 (v) f ∈ C (Ωn) This definition is to be compared with the definition of a piecewise con- tinuous function on R. Open regions Ωn are analogs of open intervals. Recall that the boundary of a region ∂Ω is smooth if it is a level set of of a function with continuous partial derivatives and a non-vanishing . A piecewise continuous function is continuous almost everywhere and at any point where it is not continuous the function can only have a jump discontinuity. In particular, a piecewise continuous function is bounded on any ball. 114 2. THE LEBESGUE INTEGRATION THEORY

A piecewise continuous function is called finite if it vanishes outside of a ball. The closure of the set in which a function f(x) =6 0 is called the of f and denoted supp f. Clearly, f is finite if and only if supp f is bounded.

The Lebesgue integral over RN . Let a real-valued function f coincide almost everywhere with the limit of a non-decreasing sequence of piece- wise continuous functions fn(x), N fn(x) ≤ fn+1(x) , ∀x ∈ R , n = 1, 2,.. such that the sequence of the Riemann integrals is bounded:

R f(x) dN x ≤ M, ∀n Z where the integral is understood in the improper sense if supports of fn are not finite. The limit

lim R fn(x)dx = f(x)dx < ∞ n→∞ Z Z of this non-decreasing bounded sequence is called the Lebesgue integral of f. The set of such functions is denoted by L+. A real function f is called Lebesgue integrable if it can be represented as the difference of two functions from L+, f = f1 − f2, f1,2 ∈ L+ and

f(x)dx = f1(x)dx − f2(x)dx . Z Z Z The set of Lebesgue integrable functions is denoted by L. Similarly to the one dimensional case, a function f is said to be from L(Ω) if the function χΩ f ∈ L, where χΩ is the characteristic function of the set Ω and, in this case, N N f d x = χΩf d x ZΩ Z Let Ω be a region and f ∈ C0(Ω). Then f ∈ L(Ω) if and only if its Riemann integral over Ω converges absolutely, that is, if and only if

lim R |f(x)| dN x< ∞ n→∞ ZΩn for some exhaustion (or regularization) {Ωn} of Ω, and, in this case,

f(x) dN x = lim R f(x) dN x. n→∞ ZΩ ZΩn A proof of this assertion is left to the reader. It is analogous to the case of continuous functions of one variables. First, the case of non- negative functions is considered for which the Lebesgue sequence is 14.LEBESGUEINTEGRAL 115

constructed as fn = χΩn f for some exhaustion {Ωn} of Ω in which Ωn are bounded. The characteristic properties of fn are easy to verify and to establish that f ∈ L+(Ω) if and only if the Riemann integral of f over Ω converges absolutely. Second, for any continuous function f, 1 ± the functions f = 2(|f| + f) are proved to be from L+(Ω) so that f = f+ − f− ∈ L(Ω).

14.5. Exercises.

1. Use the definition of the Lebesgue integral to show that the Lebesgue integral of the Dirichlet function vanishes over any interval b fD(x) dx = 0 Za 2. Let f be continuous on RN and f ∈ L(Ω). Let g be obtained from f by altering values of f on a set of measure zero. Use the definition of the Lebesgue integral to show that

g dN x = f dN x ZΩ ZΩ In other words, alterations of an integrable function on a set of measure do not change the value of the integral. Does the same property hold in the Riemann theory? If not, give an example.

3. Which of the following functions are Lebesgue integrable on R: ikx sin(x) e cos(x) 2 , , , e−x , x100e−x x x |x| 4. A function is said to be Lebesguep square integrable on Ω, or from 2 the space L2(Ω), if |f| ∈ L(Ω). Which of the functions from Problem 3 are square integrable?

5. Let f be continuous and Lebesgue integrable on RN . Show that its F[f](k)= ei(k,x) f(x) dN x Z exists for any k ∈ RN .