David M Bressoud Macalester College St Paul MN Project NExT WI October 5 2006 ―The task of the educator is to make the child‘s spirit pass again where its forefathers have gone moving ra by otj20502

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									David M. Bressoud
Macalester College, St. Paul, MN
Project NExT-WI, October 5, 2006
―The task of the educator is
to make the child‘s spirit
pass again where its
forefathers have gone,
moving rapidly through
certain stages but
suppressing none of them. In
this regard, the history of
science must be our guide.‖
              Henri Poincaré
1. Cauchy and uniform convergence
2. The Fundamental Theorem of Calculus
3. The Heine–Borel Theorem
1. Cauchy and uniform convergence
2. The Fundamental Theorem of Calculus
3. The Heine–Borel Theorem

 A Radical Approach to Real Analysis,
 2nd edition due January, 2007


 A Radical Approach to Lebesgue’s Theory of
 Integration, due December, 2007
―What Weierstrass — Cantor
— did was very good. That's
the way it had to be done.
But whether this corresponds
to what is in the depths of
our consciousness is a very    Nikolai Luzin

different question …
… I cannot but see a stark
contradiction between the
intuitively clear fundamental
formulas of the integral
calculus and the
incomparably artificial and     Nikolai Luzin

complex work of the
‗justification‘ and their
‗proofs‘.
 Cauchy, Cours d’analyse, 1821




―…explanations drawn from algebraic
technique … cannot be considered, in my
opinion, except as heuristics that will
sometimes suggest the truth, but which
accord little with the accuracy that is so
praised in the mathematical sciences.‖
     Niels Abel (1826):




―Cauchy is crazy, and there is no way of
getting along with him, even though right
now he is the only one who knows how
mathematics should be done. What he is
doing is excellent, but very confusing.‖
Cauchy, Cours d’analyse, 1821, p. 120

Theorem 1. When the terms of a series are functions of
a single variable x and are continuous with respect to
this variable in the neighborhood of a particular value
where the series converges, the sum S(x) of the series is
also, in the neighborhood of this particular value, a
continuous function of x.
Abel, 1826:

―It appears to me that this
theorem suffers exceptions.‖
x depends on n   n depends on x
“If even Cauchy can
make a mistake like this,
how am I supposed to
know what is correct?”
What is the Fundamental
Theorem of Calculus?
Why is it fundamental?
The Fundamental Theorem of Calculus (evaluation part):
       If                then

 Differentiate then Integrate = original fcn (up to constant)

The Fundamental Theorem of Calculus (antiderivative part):

      If f is continuous, then

         Integrate then Differentiate = original fcn
The Fundamental Theorem of Calculus (evaluation part):
       If                then

 Differentiate then Integrate = original fcn (up to constant)

The Fundamental Theorem of Calculus (antiderivative part):

      If f is continuous, then

         Integrate then Differentiate = original fcn
                  1820, The fundamental
                  proposition of the
                  theory of definite
                  integrals:
Siméon Poisson

    If           then
                   1820, The fundamental
                   proposition of the
                   theory of definite
                   integrals:
Siméon Poisson

    If            then

   Definite integral, defined as difference of
   antiderivatives at endpoints, is sum of
   products, f(x) times infinitesimal dx.
Cauchy, 1823, first explicit
definition of definite integral as
limit of sum of products



mentions the fact that



en route to his definition of
the indefinite integral.
Earliest reference to
Fundamental Theorem of
the Integral Calculus is by
Paul du Bois-Reymond
(1880s). Popularized in
English by E. W. Hobson:
The Theory of Functions of
a Real Variable, 1907
Granville (w/ Smith)
Differential and Integral
Calculus (starting with
1911 ed.), FTC:
definite integral can be
used to evaluate a limit
of a sum of products.
                            William A. Granville
The real FTC:
There are two distinct ways of viewing
integration:
• As a limit of a sum of products (Riemann
sum),
•As the inverse process of differentiation.
The power of calculus comes precisely from
their equivalence.
                   Riemann‘s habilitation of 1854:
                   Über die Darstellbarkeit einer
                   Function durch eine
                   trigonometrische Reihe


Purpose of Riemann integral:
1. To investigate how discontinuous a function can be
   and still be integrable. Can be discontinuous on a
   dense set of points.
2. To investigate when an unbounded function can still
   be integrable. Introduce improper integral.
     Riemann’s function:




At                         the function jumps by
The Fundamental Theorem of Calculus (antiderivative part):

      If f is continuous, then

         Integrate then Differentiate = original fcn

            This part of the FTC does not hold at
              points where f is not continuous.
The Fundamental Theorem of Calculus (evaluation part):

       If               then

 Differentiate then Integrate = original fcn (up to constant)

                      Volterra, 1881, constructed
                      function with bounded
                      derivative that is not Riemann
                      integrable.


  Vito Volterra
Perfect set: equals its set of limit points
Nowhere dense: every interval contains
subinterval with no points of the set

Non-empty, nowhere dense, perfect set
described by H.J.S. Smith, 1875
Perfect set: equals its set of limit points
Nowhere dense: every interval contains
subinterval with no points of the set

Non-empty, nowhere dense, perfect set
described by H.J.S. Smith, 1875


            Then by Vito Volterra, 1881
Perfect set: equals its set of limit points
Nowhere dense: every interval contains
subinterval with no points of the set

Non-empty, nowhere dense, perfect set
described by H.J.S. Smith, 1875


            Then by Vito Volterra, 1881

                           Finally by Georg Cantor,
                           1883
Perfect set: equals its set of limit points
Nowhere dense: every interval contains
subinterval with no points of the set

Non-empty, nowhere dense, perfect set
described by H.J.S. Smith, 1875


            Then by Vito Volterra, 1881

                           Finally by Georg Cantor,
                           1883
   Volterra’s construction:

Start with the function

Restrict to the interval [0,1/8], except find the largest value of
x on this interval at which F '(x) = 0, and keep F constant from
this value all the way to x = 1/8.
 Volterra’s construction:


To the right of x = 1/8, take the mirror image of this
function: for 1/8 < x < 1/4, and outside of [0,1/4], define
this function to be 0. Call this function       .
Now we slide this function over so that the portion that
is not identically 0 is in the interval [3/8,5/8], that
middle piece of length 1/4 taken out of the SVC set.
We do the same thing for the interval [0,1/16].
We slide one copy of        into each interval of
length 1/16 that was removed from the SVC set.
Volterra’s function, V satisfies:
1. V is differentiable at every x, V' is
   bounded.
2. For a in SVC set, V'(a) = 0, but there are
   points arbitrarily close to a where the
   derivative is +1, –1.
 V' is not Riemann integrable on [0,1]
The Fundamental Theorem of Calculus (evaluation part):

       If               then

 Differentiate then Integrate = original fcn (up to constant)

                      Volterra, 1881, constructed
                      function with bounded
                      derivative that is not Riemann
                      integrable.
                      FTC does hold if we restrict f
                      to be continuous or if we use
  Vito Volterra
                      the Lebesgue integral and F is
                      absolutely continuous.
Lessons:
1. Riemann‘s definition is not intuitively natural.
   Students think of integration as inverse of
   differentiation. Cauchy definition is easier to
   comprehend.
Lessons:
1. Riemann‘s definition is not intuitively natural.
   Students think of integration as inverse of
   differentiation. Cauchy definition is easier to
   comprehend.
2. Emphasize FTC as connecting two very different
   ways of interpreting integration. Go back to
   calling it the Fundamental Theorem of Integral
   Calculus.
Lessons:
1. Riemann‘s definition is not intuitively natural.
   Students think of integration as inverse of
   differentiation. Cauchy definition is easier to
   comprehend.
2. Emphasize FTC as connecting two very different
   ways of interpreting integration. Go back to
   calling it the Fundamental Theorem of Integral
   Calculus.
3. Need to let students know that these
   interpretations of integration really are different.
               Heine–Borel Theorem

               Any open cover of a
               closed and bounded set
               of real numbers has a
Eduard Heine   finite subcover.         Émile Borel
1821–1881                               1871–1956
                Heine–Borel Theorem

                 Any open cover of a
                 closed and bounded set
                 of real numbers has a
Eduard Heine     finite subcover.            Émile Borel
1821–1881                                    1871–1956

   Due to Lebesgue, 1904; stated and proven by
   Borel for countable covers, 1895; Heine had
   very little to do with it.
      P. Dugac, ―Sur la correspondance de Borel …‖
      Arch. Int. Hist. Sci.,1989.
           1852, Dirichlet proves that a
           continuous function on a
           closed, bounded interval is
           uniformly continuous.

The proof is very similar to Borel and
Lebesgue‘s proof of Heine–Borel.
1872, Heine reproduces this proof without
attribution to Dirichlet in ―Die Elemente der
Functionenlehre‖
               1872, Heine reproduces this proof without
               attribution to Dirichlet in ―Die Elemente der
               Functionenlehre‖



Weierstrass,1880, if a series converges
uniformly in some open neighborhood of
every point in [a,b], then it converges
uniformly over [a,b].
               1872, Heine reproduces this proof without
               attribution to Dirichlet in ―Die Elemente der
               Functionenlehre‖



Weierstrass,1880, if a series converges
uniformly in some open neighborhood of
every point in [a,b], then it converges
uniformly over [a,b].


                 Pincherle,1882, if a function is bounded in
                 some open neighborhood of every point in
                 [a,b], then it is bounded over [a,b].
Harnack, 1885, considered the question of
the ―measure‖ of an arbitrary set.
Considered and rejected the possibility of
using countable collection of open intervals.




                                                Axel Harnack
                                                 1851–1888
Harnack, 1885, considered the question of
the ―measure‖ of an arbitrary set.
Considered and rejected the possibility of
using countable collection of open intervals.




                                                Axel Harnack
                                                 1851–1888
Harnack, 1885, considered the question of
the ―measure‖ of an arbitrary set.
Considered and rejected the possibility of
using countable collection of open intervals.




                                                Axel Harnack
                                                 1851–1888

Harnack assumed that the complement of a countable union
of intervals is a countable union of intervals, in which case
the answer is YES.
Harnack, 1885, considered the question of
the ―measure‖ of an arbitrary set.
Considered and rejected the possibility of
using countable collection of open intervals.




                                                Axel Harnack
                                                 1851–1888

Harnack assumed that the complement of a countable union
of intervals is a countable union of intervals, in which case
the answer is YES.
                                  Cantor’s set: 1883
Borel, 1895 (doctoral thesis, 1894),
problem of analytic continuation
across a boundary on which lie a
countable dense set of poles
Arthur Schönflies, 1900, claimed Borel‘s result
also holds for uncountable covers, pointed out
connection to Heine‘s proof of uniform continuity.
First to call this the Heine–Borel theorem.
Arthur Schönflies, 1900, claimed Borel‘s result
also holds for uncountable covers, pointed out
connection to Heine‘s proof of uniform continuity.
First to call this the Heine–Borel theorem.


               1904, Henri Lebesgue to Borel, ―Heine says
               nothing, nothing at all, not even remotely, about
               your theorem.‖ Suggests calling it the Borel–
               Schönflies theorem. Proves the Schönflies claim
               that it is valid for uncountable covers.
Arthur Schönflies, 1900, claimed Borel‘s result
also holds for uncountable covers, pointed out
connection to Heine‘s proof of uniform continuity.
First to call this the Heine–Borel theorem.


               1904, Henri Lebesgue to Borel, ―Heine says
               nothing, nothing at all, not even remotely, about
               your theorem.‖ Suggests calling it the Borel–
               Schönflies theorem. Proves the Schönflies claim
               that it is valid for uncountable covers.

 Paul Montel and Giuseppe Vitali try to change designation
 to Borel–Lebesgue. Borel in Leçons sur la Théorie des
 Fonctions calls it the ―first fundamental theorem of measure
 theory.‖
Lessons:
1. Heine–Borel is far less intuitive than other
   equivalent definitions of completeness.
Lessons:
1. Heine–Borel is far less intuitive than other
   equivalent definitions of completeness.
2. In fact, Heine–Borel can be counter-intuitive.
Lessons:
1. Heine–Borel is far less intuitive than other
   equivalent definitions of completeness.
2. In fact, Heine–Borel can be counter-intuitive.
3. Heine–Borel lies at the root of Borel (and thus,
   Lebesgue) measure. This is the moment at
   which it is needed. Much prefer Borel‘s name:
   First Fundamental Theorem of Measure
   Theory.
This PowerPoint presentation is available at
   www.macalester.edu/~bressoud/talks
A draft of A Radical Approach to Lebesgue’s
    Theory of Integration is available at
   www.macalester.edu/~bressoud/books

								
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