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2 Integral Calculus in Several Variables

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2 Integral Calculus in Several Variables Mathematical Analysis II - Part B 2 Integral calculus in several variables In this section we approach the problem of integrating real functions of several real varia- bles. The knowledge of Riemann integral for functions of one real variable is required. 2.1 Lebesgue versus Riemann integral: motivation and a brief presentation A theory of Riemann integration for functions of several real variables (i.e. f : A R, ! with A Rn) could be constructed by adapting in a natural way the already known theory ⇢ for one variable: the role of intervals of R will be taken by the “n-dimensional intervals” I = I1 In,wheretheIj are intervals of R with extremities included or not; there ⇥···⇥ will still be the simple functions (or “step functions”, “staircase functions”) from which to start the definition of integral, the upper and lower sums, and so on. Figure 2.1: The integral of a function of two real variables. The problem of defining the integral is substantially equivalent to the one of providing a measure to subsets of Rn: one would say that E Rn is elementarily measurabile ⇢ (or measurable `ala Peano-Jordan) when its indicator function χE is Riemann-integrable, calling “measure of E” (or also “area”, “volume” in the cases n =2, 3) the value of such integral. Obvious good qualities of this theory are its simplicity and naturality; however, the drawbacks already observed in the case of one variable are still present. Let us mention three of them. Riemann-integrable functions (and, in parallel, elementarily measurable subsets) are • relatively few when one has to enlarge its scope beyond continuous functions, or rat- her beyond the functions having a “reasonable number of singular points”.(13) Na- (13)This is what happens since more than one century in the progress of exact sciences, as new important Corrado Marastoni 29 Mathematical Analysis II - Part B mely, on a compact n-dimensional interval a Riemann-integrable function must be bounded, and continuous “almost everywhere”:(14) for example, this fact says that the Dirichlet function χ (which is discontinuous in all points [0, 1]) cannot be Q [0,1] Riemann-integrable. \ The Riemann integral has no satisfactory interaction with the limit.Givenasequence • of integrable functions fn which converges to a function f in a sufficiently weak way (e.g. pointwise, or even less), we would like also f to be integrable and that it could be possible to “pass to limit under the sign of integral”, i.e. that f =lim fn. Now, in such a generality, this request is rather optimistic, because a weak convergence cannot R R control “mass dispersion” phenomena in the domain, as the following example shows. n n (n 1) Example. Let us define fn : R R as 2 on the interval [2− , 2− − ], and 0 elsewhere. Any of ! the fn is Riemann-integrable with fn = 1, and the sequence (fn) converges pointwise to the zero function f 0, which is obviously Riemann-integrable as well; but f =0=lim fn =1. ⌘ R 6 R R To prevent such phenomena, a reasonable request is that this should be true at least for sequences “dominated” by a fixed integrable function, i.e. such that there exists an integrable function g such that fn g for any n N. However, for the Riemann | | 2 integral also this is not enough: in fact the convergence should be uniform, because pointwise convergence could not work. Example. Let q : N Q be any bijection (we know that Q is countable), and define fn : R R ! ! as 1 onlu on the rationals qm with m n, and 0 elsewhere: any of the fn is Riemann-integrable (it has finitely many discontinuity points) with fn =0,butthesequence(fn) converges pointwise (not uniformly!) to Dirichlet function f = χ [0,1] , which is not Riemann-integrable. Q\ R The Riemann integral is linked to the properties of real numbers. Riemann’s costruction • depends a lot on the properties of real numbers (in particular on the total order), and the extendibility to the case of real functions defined on more general sets than Rn appears to be difficult. These reasons, among some others, suggested at the end of XIXth century to look for a more e↵ective integration theory, being prepared to pay a price in terms of simplicity but without loosing too much of its naturality: among various attempts, the one by Henri tools (for example, Fourier series) could generate highly irregular functions. (14)The precise meaning of this “almost everywhere” is —as we shall see later— “everywhere excepted a null set”, where “null set” means a set of points whose Lebesgue measure is zero: the exact statement is n that a bounded function with compact support f : R R is Riemann-integrable if and only if the set of its ! discontinuity points is a null set. The Lebesgue measure, as we shall see, is much larger than the elementary measure of Peano-Jordan. For example, all countable sets are Lebesgue-measurable (with measure zero), 1 while they could be not elementarily measurable: for example E = n : n N is elementarily measurable 1 1 { 12 } (if sn : R R is the simple function with values 1 on [0, n ] n 1 ,..., 2 , 1 and zero elsewhere, it holds ! 1 [{ − } χE sn and hence sn = is a upper sum for χE ), while the Dirichlet set Q [0, 1] is not. A classical R n \ example of Riemann-integrable function whose discontinuity set is Lebesgue-null but not elementarily R p measurable is f :[0, 1] R defined as 0 on [0, 1] Q and as 1 on Q [0, 1] with p, q 0 coprime, ! \ q q 2 \ ≥ whose discontinuity set is the Dirichlet set Q [0, 1]. \ Corrado Marastoni 30 Mathematical Analysis II - Part B Lebesgue’s (1875-1941) eventually turned out to be the most successful. Let us discuss why by making reference to the just mentioned Riemann’s drawbacks. When compared to Riemann theory, Lebesgue’s considerably enlarges the family of • measurable sets (and, in parallel, the family of measurable functions, possibly integra- ble), so that it is very complicated to describe non Lebesgue-measurable subsets of Rn; in particular, the Dirichlet function χ is Lebesgue-integrable with integral zero. Q [0,1] The role played by continuity in Riemann\ integral is played here by the property of “measurability” which is extremely weak and in practice satified all the time. Lebes- gue’s construction, which nowadays keeps having a large agreement(15), is in a certain sense the result of a process of completion of Riemann’s, similar to the construction of real numbers starting from rationals. The Lebesgue integral works much better under limit with respect to Riemann’s: if • we have a sequence of Lebesgue-integrable functions fn which converges pointwise “almost everywhere”(16) to a function f, in two very frequent cases (a monotone or a dominated sequence) also f is Lebesgue-integrable and f =lim fn. In particular the example with f = χ [0,1] is fixed (the sequence fn converges monotonically to f, Q\ R R which is Lebesgue-integrable and 0 = χ =lim f ), while the example with Q [0,1] n the “mass dispersion” keeps resisting (actually\ the sequence f converges pointwise R R n to f = 0 but neither in a monotone nor in a dominated way). Lebesgue theory is largely independent from particular properties (topological or not) • of the domain, and hence it naturally suits to real functions defined on any domain: in fact, to present the theory in full generality does not create more difficulties with respect to Rn (on the contrary, it rather helps to better clarify the situation). Anyway, we should point out that a large part of the results which are valid for • Lebesgue theory are valid also for Riemann’s: but usually the hypotheses in the latter are some heavier (and, after all, quite unessential). For what has been said, it should appear convenient to make reference to Lebesgue integral from now on. Let us briefly recall the main aspects of this theory, without proofs. Let X be any set, and let us plan to introduce a measure µ of the subsets of X: in other words, to a A X we want to associate a measure µ(A), that we expect to be a number ⇢ 0, possibly + . How can we do? The first requirement would be, at one side, to be able ≥ 1 to measure as many subsets of X as possible; but, on the other side, the measure should satisfy some natural properties, that we now list. (15)We should however say that, while Lebesgue theory fixes many problems, some unexpected others sin x appear: for example x is not Lebesgue-integrable on R although it has a generalized Riemann-integral (called “Dirichlet integral”); in fact it is not absolutely Riemann-integrable. (16)i.e., as said above, “excepted a null set”: it is clearly a rather weak notion of convergence. Corrado Marastoni 31 Mathematical Analysis II - Part B – Enough stability for the set-theoretical operations . We aim that the family of µ- measurable subsets, beyond than being rich of elements, would be stable under the main set-theoretical operations, in particular under the union and the intersection (even better, under countable union and intersection) and under complementation in X. In other words: if A, B X are µ-measurable, we aim that also A B, A B and ⇢ [ \ {X A = X A be so; and that, if An is a countable family of µ-measurable subsets, \ (17) also An and An be so.
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