Integration in lattice spaces

Thesis Overview

<p> Introduction to Integration Theory<br>1.1. Riemann-Stieltjes Integration<br>Definition of the Riemann-Stieltjes integral on a compact set<br>Consider an arbitrary function f : [a; b] ! R.<br>The Riemann-Stieltjes integral of f on [a; b] associated with F, if it exists,<br>is denoted by:<br>I =<br>Z b<br>a<br>f(x) dF(x)<br>In establishing the existence of the Riemann-Stieltjes integral of a function,<br>we need the function to be bounded.<br>Next, we define the Riemann-Stieltjes sums. To do so, for each n 1, we<br>divide [a; b] into l(n) sub-intervals (l 1).<br>Let n be a subdivision of [a; b] that divides[a; b] into l(n) sub-intervals.<br>So,<br>]a; b] =<br>l(Xn)ô€€€1<br>i=0<br>]xi;n; xi+1;n];<br>where a = x0;n &lt; x1;n &lt; ::: &lt; xl(n);n = b:<br>5<br>6 1. INTRODUCTION TO INTEGRATION THEORY<br>The modulus of the subdivision n is defined by:<br>m(n) = max<br>0il(n)ô€€€1<br>(xi+1;n ô€€€ xi;n)<br>Then, in each sub-interval ]xi;n; xi+1;n], we pick an arbitrary point ci;n,<br>we therefore have the arbitrary sequence (cn)n1 where, cn = (ci;n)1il(n)ô€€€1.<br>we now define a sequence of Riemann-Stieltjes sum associated to the subdivision<br>n and the vector cn in the form:<br>(1.1.1) Sn(f; F; a; b; n; cn) =<br>l(Xn)ô€€€1<br>i=0<br>f(ci;n)(F(xi+1;n) ô€€€ F(xi;n))<br>in short, Sn(n; cn)<br>Definition 1.1. A bounded function f is Riemann-Stieltjes integrable<br>with respect to F if there exists a real number I such that any sequence<br>of Riemann-Stieltjes sums Sn(n; cn) converges to I as n ! 1 whenever<br>m(n) ! 0 as n ! 1.<br>The number I is called the Riemann-Stieltjes integral of f on [a; b]<br>Now, in particular, if F(x) = x; x 2 R, I is called the Riemann Integral of f<br>over [a; b] and the sum in formula 1.1.1 is simply called the Riemann Sum.<br>For the sake of a later use, Let us introduce an important notion called<br>”‘Bounded Variation Functions”’.<br>1.2. BOUNDED VARIATION FUNCTIONS 7<br>1.2. Bounded Variation Functions<br>Consider a function F : [a; b] ! R.<br>We define by P(a; b) the class of all partition of the interval [a; b] of the<br>form:<br>(1.2.1) = (a = x0 &lt; x1 &lt; ::: &lt; xp = b); p 1<br>To each 2 P(a; b) represented as in formula 1.2.1, we associate the variation<br>of F over define by:<br>VF (; a; b) =<br>Xp<br>j=i<br>jF(xj+1) ô€€€ F(xj)j<br>The total variation of F over [a; b] is defined by:<br>VF (a; b) = sup<br>2P(a;b)<br>VF (; a; b)<br>Definition 1.2. A function F is said to be of bounded variation if and<br>only if its total bounded variation over [a; b] is finite, that is:<br>0 VF (a; b) = sup<br>2P(a;b)<br>VF (; a; b)<br>Example 1.3. (1) Any non-decreasing function F : [a; b] ! R is of<br>bounded variation.<br>We have, for all 2 P, VF (; a; b) = F(b) ô€€€ F(a), So :<br>VF (a; b) = F(b) ô€€€ F(a) &lt; +1<br>8 1. INTRODUCTION TO INTEGRATION THEORY<br>(2) Any non-increasing function F : [a; b] ! R is of bounded variation.<br>We have, for all 2 P, VF (; a; b) = F(a) ô€€€ F(b), So :<br>VF (a; b) = F(a) ô€€€ F(b) &lt; +1<br>(3) Any continuously differentiable (C1) function F : [a; b] ! R is of<br>bounded variation.<br>In fact, since F0 2 C[a; b], then M := sup<br>x2[a;b]<br>jF0(x)j &lt; +1 Now, for all<br>2 P(a; b), by the Mean Value Theorem, 8 j = 1; :::; p; 9 2 [0; 1] such<br>that:<br>F(xj) ô€€€ F(xjô€€€1) = (xj ô€€€ xjô€€€1)F0(xjô€€€1 + j(xj ô€€€ xjô€€€1));<br>So,<br>VF (; a; b) =<br>Xp<br>j=1<br>(xj ô€€€ xjô€€€1)jF0(xjô€€€1 + j(xj ô€€€ xjô€€€1))j<br>M(b ô€€€ a)<br>Therefore,<br>VF (a; b) = sup<br>2P<br>(a; b)VF (; a; b) M(b ô€€€ a) &lt; +1<br>Lemma 1.4. Any bounded variation function on [a; b] is a difference of two<br>non-decreasing function.<br>Now, consider a continuous function f : [a; b] ! R. Our interest here is<br>to show the existence of the Riemann-Stieltjes integral of f. f being so<br>1.2. BOUNDED VARIATION FUNCTIONS 9<br>smooth, we should at least expect, for a strong theory of integration, f to<br>be Riemann-Stieltjes integrable.<br>However, for what function F can we define the Riemann-Stieltjes integral<br>of f.<br>Theorem 1.5. If F is of bounded variation, every continuous function<br>on [a; b] is integrable, i.e, has a Riemann-Stieltjes integral I denoted by:<br>I =<br>Z b<br>a<br>f(x) dF(x)<br>The Riemann-Stieltjes integration is limited. In fact, we started the construction<br>by first assuming that our function f is bounded and is defined<br>on the interval of the form [a; b]. Moreover, we also considered different<br>parameters in establishing the Riemann Sum.<br>For example, Let F(x) = x. So to determine the Riemann integral of f :<br>[a; b] ! R, bounded, we need to compute the Riemann Sums. In fact, in<br>the process of computing the Riemann sums, for a fixed n, we are technically<br>computing sum of areas of small rectangles of width w = xiô€€€xiô€€€1; 1<br>i l(n).<br>However, to approximate the lengths of triangle, we arbitrarily choose a<br>point ci between xiô€€€1 and xi and we use the image f(ci) of the point ci, in<br>computing the areas of those triangle. That is, we can choose any ci in<br>]xiô€€€1; xi].<br>For our approximation to make sense, we need to have that for any two<br>points arbitrarily chosen in the sub-interval ]xiô€€€1; xi], the images of those<br>points are not far from one another in terms of value. In order words, the<br>10 1. INTRODUCTION TO INTEGRATION THEORY<br>function f should be continuous.<br>However, in real-life situation, we hardly meet smooth functions. Therefore,<br>we make use of the Lebesgue integration which mainly requires only<br>measurality of functions.<br>The illustration is given below.<br>Figure 1. Geometric Interpretation of Riemann integration where we arbitrarily<br>chose our ci to be xi+1.<br>1.3. LEBESGUE INTEGRATION 11<br>1.3. Lebesgue Integration<br>1.3.1. Distribution function on R.<br>Definition 1.6. A function F : R ! R is called a distribution function if<br>and only if:<br>(i) F is right continuous<br>(ii) F assigns to intervals non-negative lengths i.e 8 a b, F(b) ô€€€ F(a) 0<br>1.3.2. Lebesgue-Stieltjes measure associated to F. We construct the<br>Lebesgue-Stieltjes measure on (R; B(R)).<br>B(R) = (S)<br>where S = f]a; b]; a &lt; bg is a semi algebra.<br>Define:<br>F : S ! R+<br>]a; b] ! F (]a; b]) = F(b) ô€€€ F(a)<br>F is called the Lebesgue-Stieltjes measure.<br>If F(x) = x; F = is the Lebesgue measure on R<br>1.3.3. The Lebesgue-Stieltjes Integral. Let F : R ! R be a distribution<br>function.<br>For f, measurable, the Lebesgue-Stieltjes integral of f with respect to the<br>measure F is denoted as:<br>I =<br>Z<br>f(x) dF (x)<br>12 1. INTRODUCTION TO INTEGRATION THEORY<br>The construction of this type of integral, depending on some properties of<br>f, is given in chapter 3.<br>In fact, this thesis is mainly about the integration of measurable mappings<br>with respect to measure.<br>Also, for the coherence in the theory of integration, it is not a surprise<br>that the Riemann-Stieltjes integration and the Lebesgue-Stieltjes integration<br>sometimes coincide.<br>Example 1.7. (1) Let f : [a; b] ! R, a &lt; b,f bounded.<br>f is Riemann integrable if and only if f is ô€€€a:e continuous; and the<br>Riemann and the Lebesgue integrals coincide.<br>(2) Any Riemann integral on the compact set [a; b] is a Lebesgue integral<br>on [a; b]<br>Furthermore the notion of Lebesgue-Stieltjes integration is broader than<br>the notion of Riemann-Stieltjes integration, because all Riemann-Stieltjes<br>integrable functions are Lebesgue-Stieltjes integrable but not all Lebesgue-<br>Stieltjes integrable functions are Riemann integrable.<br>Example 1.8. f = 1[a;b]<br>T<br>Q is Lebesgue integrable but not Riemann integrable.<br>This chapter is a brief introduction to the theory of integration. All types<br>of integration have not been discussed. Here, we only introduced the<br>Riemann-Stieltjes integration and addressed a broader type of integration<br>1.3. LEBESGUE INTEGRATION 13<br>called the Lebesgue integration.<br>In fact, the Lebesgue-Stietjes integration is simply the integration of realvalued<br>measurable mappings with respect to the Lebesgue-Stieltjes measure.<br>In coming chapters, we will discuss the integration of measurable functions<br>with respect to any arbitrary measure on some specific cases. Depending<br>on the space, we put a finiteness condition on the <br></p>