College Math Teaching

January 17, 2011

Integration: Riemann Integration, Limitations and Lebesgue’s Idea

What about integration? Here we will see what Lebesgue integration is about, how it differs from Riemann integration and why we need to learn about the algebra of measurable sets.

Brief review of Riemann Integration

Remember that the idea was as follows: we limit ourselves to bounded functions. suppose we want to compute \int_{a}^{b}f(x)dx. We partitioned the interval [a,b] into several subintervals:

a=x_{0}<x_{1}<x_{2}...<x_{n-1}<x_{n}=b. Let m_{i}=\inf f(\xi ),\xi \in \lbrack x_{i-1},x_{i}] and let M_{i}=\sup f(\omega ),\omega \in \lbrack x_{i-1},x_{i}]. Let \Delta x_{i}=x_{i}-x_{i-1}. Call this partition P.

Then L_{P}=\sum_{j=1}^{n}m_{j}\Delta x_{j} and U_{P}=\sum_{j=1}^{n}M_{j}\Delta x_{j} are called the lower sums and upper sums for f with respect
to the partition P.

One proves theorems such as if Q is a refinement of partition P then L_{P}\leq L_{Q} and U_{Q}\leq U_{P} (that is, as you make the refinement finer…with smaller intervals, the lower sums go up (or stay the same) and the upper sums go down (or stay the same) and then one can define U to the
be infimum (greatest lower bound) of all of the possible upper sums and L to the the supremum (least upper bound) of all of the possible lower sums. If U=L we then declare that to be the (Riemann) integral of f over [a,b].

Note that this puts some restrictions on functions that can be integrated; for example f being unbounded, say from above, on a finite interval will prevent upper sums from being finite. Or, if there is some dense subset of [a,b] for which f obtains values that are a set distance away from the the values that f attains on the compliment of that subset, the upper and lower sums will never converge to a single value. So this not only puts restrictions on which functions have a Riemann integral, but it also precludes some “reasonable sounding” convergence theorems from being true.

For example, suppose we enumerate the rational numbers by q_{1},q_{2},...q_{k}... and define f_{1}(x)=\left\{\begin{array}{c}1,x\neq q_{1} \\ 0,x=q_{1}\end{array}\right. and then inductively define f_{k}(x)=\left\{\begin{array}{c}1,x\notin \{q_{1},q_{2},..q_{k}\} \\ 0,x\in \{q_{1},q_{2},..q_{k}\}\end{array}\right.  . Then f_{k}\rightarrow f=\left\{\begin{array}{c}1,x\notin \{q_{1},q_{2},..q_{k}....\} \\ 0,x\in \{q_{1},q_{2},..q_{k},...\}\end{array}\right. and for each k, \int_{0}^{1}f_{k}(x)dx=1 but f, the limit function, is not Riemann integrable.

So, there are a couple of things to note here:

1. The Riemann integral involves partitioning the interval to be integrated over without regards to the function being integrated at all; that is, if you were doing \int_{0}^{1}e^{\sqrt{x}}dx or \int_{0}^{1}\sin (x^{2})dx you wouldn’t partition [0,1] any differently.

2. The elements of any partition of the Riemann integral are intervals of finite length.

The Lebesgue integral changes these two features;

1. We’ll use information about the function being integrated to help us select partitions and

2. The elements of our partition need not be intervals of finite length; they just need to be measurable sets.

For example, suppose we wish to compute \int_{0}^{1}4x-x^{2}dx by using a Lebesgue integral.

Partition the range of f into 4 subintervals:

Y_{1}=0\leq y<.25,

Y_{2}=.25\leq y<.5,

Y_{3}=.5\leq y<.75,

Y_{4}=.75\leq y\leq 1.

Now consider the inverse image of these subintervals and label these:

E_{1}=f^{-1}(Y_{1})=[0,\frac{1}{2}-\frac{1}{4}\sqrt{3})\cup (\frac{1}{2}+\frac{1}{4}\sqrt{3},1]

E_{2}=f^{-1}(Y_{2})=[\frac{1}{2}-\frac{1}{4}\sqrt{3},\frac{1}{2}-\frac{1}{2}\sqrt{\frac{1}{2}})\cup \lbrack \frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}},\frac{1}{2}+\frac{1}{4}\sqrt{3})

E_{3}=f^{-1}(Y_{3})=[\frac{1}{2}-\frac{1}{2}\sqrt{\frac{1}{2}},\frac{1}{4})\cup \lbrack \frac{3}{4},\frac{1}{2}+\frac{1}{2}\sqrt{\frac{1}{2}})

E_{4}=f^{-1}(Y_{4})=[\frac{1}{4},\frac{3}{4})

Then we form something similar to upper and lower sums. Recall the measure of an interval is just its length.

So we obtain something like an upper sum:

U=\frac{1}{4}m(E_{1})+\frac{1}{2}m(E_{2})+\frac{3}{4}m(E_{3})+1m(E_{4})

and a lower sum as well:

L=0m(E_{1})+\frac{1}{4}m(E_{2})+\frac{1}{2}m(E_{3})+\frac{3}{4}m(E_{4})

See the above figure for an illustration of an upper sum.

Then we proceed by refining the partitions of our range; for this to work we need for the inverse image of the partitions of the range to be measurable sets; this is why we need theorems about what constitues a measureable set.

A measurable function is one whose inverse images (or partitions of its range into intervals) are measurable sets.

The Lebesgue integral can be defined as either the infimum of all the upper sums or the supremum of all of the lower sums.

If one wants to see how this works, try doing this for \int_{0}^{1}g(x)dx where
g(x)=\left\{ \begin{array}{c}1,x\notin Q \\ \frac{1}{p},x=\frac{p}{q}\end{array}\right. where Q is the rationals and \frac{p}{q} is in lowest terms.

Then the subinterval which includes 1 in any partition of the range will have an inverse image with measure 1 whereas all subintervals whose upper bounds are strictly less than 1 will have measure zero. Hence it follows that \int_{0}^{1}g(x)dx=1 though g is not Riemann integrable.

Of course, this has been sketchy and we haven’t covered which types of sets are measurable. We’ll also discuss some convergence theorems as well.

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5 Comments »

  1. […] Integration and measure I Lebesgue Integration and measure 2 Lebesgue Integration and measure 3 Lebesgue Integration and measure […]

    Pingback by 18 January 2011: posting « blueollie — January 18, 2011 @ 12:45 pm

  2. if Q is refinement of P then L(P)<=L(Q).

    Comment by Jyoti — February 24, 2015 @ 11:28 am

    • thank you…..that was a typo. lower sums get larger, upper sums get smaller.

      Comment by blueollie — February 24, 2015 @ 12:56 pm

  3. here indeed as

    Comment by dedusuiu — February 22, 2018 @ 4:06 pm

  4. also we see as say prof dr mircea orasanu and prof horia orasanu as followed
    LEBESGUE AND INTEGRAL OF LAGRANGIAN

    ABSTRACT
    In an attempt to make logical proofs rigorous, logicians at the end of the nineteenth century and the beginning of the twentieth developed axiom systems for formal logic in the spirit of Euclid’s axiom system for geometry
    1 INTRODUCTION
    Inference rules can be written as {1 ,  ,n} ⊢ , indicating that statement  can be inferred from the set of statements 1 ,  ,n. A proof in an axiom system S of is a sequence of assertions, starting with a set of assertions  and ending with a assertion , where each assertion  in the sequence after  is either an axiom, or is derived from a subset of the assertions in the sequence before  by the application of some inference rule. If there exists a formal proof in S from  to , we write  ⊢S

    OPTIMIZARI CU LEGATURI NEOLONOME
    Prof. Horia Orasanu .
    CAP1 LAGRANGIAN SI NON HOLONOMIE
    Funcția H este cunoscută ca Hamiltonian sau funcția energetică, iar mulțimea simplectică se numește spațiul fazelor. Hamiltonianul induce un câmp vectorial special peste o mulțime simplectică, cunoscut drept câmp vectorial simplectic.
    In acest caz notiunea si motivarea acestei probleme este posibila prin cele prezentate
    Aici avem ca in cazul optimizarii constrangerilor neolonome ,prin relatiile matematice
    De forma [1] cuațiile lui Hamilton sunt atractive având în vedere simplitatea și simetria lor. Ele au fost analizate din toate punctele de vedere imaginabile, de la mecanica fundamentală la geometria spațiilor vectoriale. Se cunosc o serie întreagă de soluții ale acestor ecuații, dar soluția generală exactă a ecuațiilor de mișcare pentru sisteme cu mai mult de două corpuri nu se cunoaște încă. Găsirea integralelor prime, adică a mărimilor care se conservă, joacă un rol important în găsirea soluțiilor sistemului, sau al informațiilor despre natura lor. Modelele cu un număr infinit de grade de libertate, evident sunt mult mai complicate, dar o arie interesantă de cercetare este studiul sisteme

    S = 2

    Astfel ca se pot aplica si alte modalitati,precum teoria ecuatiilor diferentiale de o variabi
    La complexa [2]
    ,
    Astfel ca in acest caz putem sa ne referim la portiuni de frontiere adaugate ,in care vorticitatea
    Asociata unui vortex sunt mici [3]
    Ecuațiile lui Hamilton sunt ecuații diferențiale de ordinul întâi, ele fiind mai ușor de rezolvat decât ecuațiile lui Lagrange, care sunt de ordinul doi. Cu toate acestea, pași care conduc la ecuațiile de mișcare sunt mai costisitori decât în mecanica lui Lagrange – începând cu coordonatele generalizate și Lagrangianul, trebuie să calculăm hamiltonianul exprimând fiecare viteză generalizată în termenii coordonatelor generalizate, pe care o vom înlocui în hamiltonian. În final, vom obține aceeași soluție ca în mecanica lui Lagrange sau folosind legile de mișcare Newtoniene. and
    n coordonate Carteziene, impulsul generalizat corespunde exact impulsului. În coordonate polare, impulsul generalizat corespunde momentului unghiular, iar prin alegerea unei coordonate generalizate oarecare, este posibil să nu obținem o interpretare intuitivă fizică a coordonatei canonice.
    Un lucru care nu este prea evident în acestă formulare dependentă de coordonată, faptul că, diferite coordonate generalizate nu sunt altceva decât sisteme de coordonate diferite ale aceluiași spațiu vectorial.
    Hamiltonianul este de fapt transformarea Legendre a Lagrangianului:

    De aici decucem imediat ca avem o alta forma a ecuatiilor oscilatiilor cu constrangeri neo
    Lonome care sunt de forma [4]
    J (y) =

    Intrucat exista relatiile echivalente [5]

    y1 = y(x1) and y2 = y(x2).
    Aceste rezultate pot fi extinse la oscilatii in cazul fluidelor grele nemiscibile ,date de ecua
    Tia lui Buseman si Boussinesq

    ceasta se numește teorema lui Liouville: Fiecare funcție netedă G peste o mulțime simplectică generează o familie uniparametrică de simplectomorfisme, iar dacă { G, H } = 0, atunci G se conservă, iar simplectomorfismele sunt transformări simetrice.
    Hamiltonianul poate avea multe cantități Gi care se conservă. Dacă mulțimea simplectică are dimensiunea 2n și dacă există n cantități Gi independente funcțional care se conservă, fiind în involuție (adică, { Gi, Gj } = 0), atunci Hamiltonianul este integrabil în sensul lui Liouville. Teorema Liouvile-Arnol’d afirmă că, local orice Hamiltonian integrabil în sensul lui Liouville poate fi transformat printr-un simplectomorfism într-un Hamiltonian cu cantitățile Gi conservate sub forma coordonatelor, iar noile coordonate se numesc coordonate unghi-acțiune. Hamiltonianul transformat depinde numai de Gi, și astfel ecuația de mișcare capătă forma simplă:

    Si aceasta metoda este analoaga teoriei undelor de amplitudine mica pe o suprafata a unu
    I fluid greu, ceea ce presupune o toplologie ,care fiind impusa de potentialul vitezei notat

    Si astfel avem functia care satisface la o conditie Lipschitz dar in origine nu are derivata s
    Si in alte puncte nu are singularitati

    Deci in continuare putem sa vedem care este comportamentul Lagrangeanului in capetele
    Unui interval. In acest caz putem formula teorema
    Toate aceste circumstante sunt associate cu absorbtia care apare la caldura latenta ,ce
    Apare la deplasarea unei interfete sau a mai multora , intre care mai multe faze.

    ,

    Deci rezultatele numerice pentru schema si problemele neolonomice in cazul problemelor cu cavitati
    Pot fi comparate cu existent solutiilor din punct de vedere fizic.

    Deasemenea problema legaturilor neolonome se aplica si in cazul situatiilor ecologice
    Fara sa apara reactii de vartej sau vortex ,si aici adaugam ca in timp ce se ia in diverse
    Lucrari
    ,

    . References
    [1] Y. Yamamoto, and X. Yun, “Coordinated obstacle avoidance of a mobile manipulator”, Proceedings of the IEEE Conference on Robotics and Automation, pp.2255-2260, 1995.
    [2] Y. Yamamoto, and X. Yun, “Unified analysis on mobility and manipulability of mobile manipulators”, Proceedings of the IEEE Conference on Robotics and Automation, pp.1200-1206, 1999.
    [3] R. Colbaugh, “Adaptive stabilization of mobile manipulators”, Proc. of the Amer. Controls Conf., pp. 1-5, 1998.
    [4] A. M. Bloch, M. Reyhanoglu, and N. H. McClamroch, “Control and stabilization of nonholonomic caplygin dynamic systems”, Proc. of the IEEE Conference on Decision and Control, pp. 1127-1132, December 1991.
    [5] S. Jagannathan, F. L. Lewis and K. Liu, “Motion control and obstacle avoidance of mobile robot with an onboard manipulator”, Journal of Intelligent Manufacturing Systems, vol.5, pp. 287-302, 1994.
    [6] S. Jagannathan, S. Q. Zhu and F. L. Lewis, “Path planning and control of a mobile base with nonholonomic constraints”, Robotica, vol. 12, part 6, pp. 529-540, 1994.

    Comment by dedusuiu — February 22, 2018 @ 4:23 pm


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