Familiarizing students with definition of Lebesgue integral - examples of calculation directly from its definition using Mathematica

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1 Familiarizing students with definition of Lebesgue integral - examples of calculation directly from its definition using Mathematica Włodzimierz Wojas, Jan Krupa Warsaw University of Life Sciences (SGGW), Poland, wlodzimierz_wojas@sggw.pl Warsaw University of Life Sciences (SGGW), Poland, jan_krupa@sggw.pl Young man, in mathematics you don t understand things. You just get used to them John von Neumann In popular books of calculus, for example [, ], we can find many examples of Riemann integral calculated directly from its definition. The aim of these examples is to familiarize students with the definition of Riemann integral. But we cannot find analogical examples for Lebesgue integral. In this article, with similar aim but for Lebesgue integral definition, we present the following examples of calculation directly from its definition: b a exp(x) dm(x), π x dm(x), x k dm(x), π/ sinxdm(x), ln( r cosx+r )dm(x), where dm(x) denotes the Lebesgue measure on the real line. We calculate sums, limits and plot graphs of needed simple functions using Mathematica. The two following definitions of Lebesgue integral are used in this article: Let (R,M,m) be measure space, where M is σ algebra of Lebesgue measurable subsets in R, and m- Lebesgue measure on R. Let f : R R be measurable nonnegative function (we ve omitted the definition of Lebesgue integral for simple real measurable functions). Definition. (See [3, 5, 6, 7, 8]) { f dm(x) = sdm(x) : s f,s simple measurable function. () Definition. (See [4, 9, ]) Let s n be nondecreasing sequence of nonnegative simple measurable functions such that lim s n (x) = f (x) for every x R. Then: f dm(x) = lim s n dm(x). ()

2 Example. Let consider the function: f (x) = sin x, x [, π/). For the rest of this example we will restrict our consideration to x [,π/). We will calculate π/ sinxdm(x) applying directly definition. Consider n ( ) k s n (x) = sin k= n+ π χ [ ) k k+ (x), for x [,π/), n =,,... n+ π, n+ π n ( ) k and s n (x) = s n = sin k= n+ π χ [ ) k n+ π, k (x), for x [,π/), n =,,... n+ π Using Wolfram Mathematica we get the following Figures, : y sin x s x s x s x s x x Figure : Graphs of functions f, s,s. We can see that s (x) s (x) for x [,π/).

3 y sin x s x s 3 x s x s 3 x x Figure : Graphs of functions f, s,s 3. We can see that s (x) s 3 (x) for x [,π/). It is clear that s n, s n are sequences of nonnegative simple measurable functions and that s n f and s n f on [,π/) for all n =,,... Using Wolfram Mathematica we get: Listing : Mathematica code: In[]=Sum[Sin[Pi k/^(n+)], {k,, ^n ] Pi/^(n+)//Simplify Out[]=^( n)pi( +Cot[^( n)pi]) In[]=Limit[%,n >Innity] Out[]= So a n = Similarly n s n dm)x) = k= sin kπ n+ n+ π = n π( + cot( n π)) (3) Listing : Mathematica code: In[3]=Sum[Sin[Pi k/^(n+)], {k,, ^n] Pi/^(n+)//Simplify Out[3]=^( n)pi(+cot[^( n)pi]) 3

4 In[4]=Limit[%,n >Innity] Out[4]= So ā n = s n dm(x) = n k= sin kπ n+ n+ π = n π( + cot( n π)) (4) n n+ sin Of course we could use the following formulae: sin(kx) = xsin n x k= sin x and sinx lim = instead of the code in Listings and to get the results in formulae x x (3) and (4). Using formulae (3) and (4), basic properties of least upper, greatest lower bounds and of Lebesgue integral of simple measurable functions we will prove in our talk that: { sdm(x) : s f,s simple measurable function (5) and { sdm(x) : s f,s simple measurable function. (6) Inequalities (5) and (6) give { sdm(x) : s f,s simple measurable function =, which means that f dm(x) = sinxdm(x) =. Let calculate sinxdm(x) applying directly definition. We can see that s n (x) s n+ (x) for x [,π/) and for all n =,,... In Figures and we can see that s (x) s (x) and s (x) s 3 (x) for x [,π/). W can also see that lim s n (x) = sin(x) for all x [,π/). So s n is nondecreasing sequence of nonnegative simple measurable functions and s n converges pointwiseto f. So by formula (3) and directly by definition we get sinxdm(x) = f dm(x) = lim s n dm(x) = lim a n =. 4

5 References [] Tom M. Apostol, Calculus, Volume, One-Variable Calculus with an Introduction to Linear Algebra, nd ed., Addison-Wesley Publishing Company, (99) [] G. M. Fichtenholz, Differential and Integral Calculus, Fizmatgiz, (958) [3] W. Rudin, Principles of Mathematical Analysis, 3 rd ed., McGraw-Hill Education, (973) [4] Charalambos D. Aliprantis, Owen Burkinshaw, Principles of Real Analysis, 3 rd ed., Academic Press, (998) [5] Robert G. Bartle, The Elements of Integration and Lebesgue Measure, Wiley-Interscience, (995) [6] Frank Jones, Lebesgue Integration On Euclidean Space, Jones & Bartlett Learning, () [7] Andrew Browder, Mathematical Analysis An Introduction, nd ed., Springer, () [8] Gerald B. Folland, Real Analysis Modern Technique, nd ed., Wiley, (7) [9] W. Kołodziej, Mathematical Analysis, (in polish), Polish Scientific Publishers PWN, Warsaw () [] R. Sikorski, Differential and Integral Calculus. Functions of several variables, (in polish), Polish Scientific Publishers PWN, Warsaw (977) [] H. Ruskeepa Mathematica Navigator: Graphics and Methods of applied Mathematics. Academic Press, Boston (5) [] S. Wolfram The Mathematica Book. Wolfram Media/ Cambridge University Press (996) 5