Calculus

When I am asked to give a short description of the key issue of mathematical work, my answer is: Mathematicians explore and analyze the rules of counting.

What are the essentials of counting?

The integer numbers are generated by a single element. We call it "one". The art of counting is to find a name for the successor of each number.

The most popular decimal System features ten ciphers, which allows to use the fingers as a simple device for counting and computing.

We want to make sure that for a given an arbitrary integer x there is a successor.

The Italian mathematician Peano posted some axioms to describe the integer numbers. His method was based on the principles of induction and recursion. In both cases you need a starting rule and a transcription for progressing from an arbitrary number n to n+1.

The faculty of a number w is defined as the product of all numbers n, which are less or equal to w. The faculty of one is equal to one obviously.

 Everybody may now have an idea, in which way the function Faculty has to be computed. Actually there are a lot of different methods around. We wil use the folllwing notation to commnicate and to document such a procedure.

Faculty (n) {
var ; begin w:=w*n;n:=n-1;if n>0 then call Faculty(n) end }

Faculty keeps on calling itself, until the  parameter n has been decremented to zero. In each cycle the variable w is multiplied by n acording to the definition of faculty numbers.

In order to understand how this recursive routine works the reader should choose a number  n and do all the computations manually.

The first theory of sets from a naive point of view was developed by the German mathematician Cantor.

A set was defined as a container for objects called elements, which could be listed explicitly by name, or implicitly by a logical statement. Cantor used the Greek letter epsilon to express the relationship that an object A is contained as an element in the set B: A € B.

The British philosopher and mathematician Russell defined a set R having all objects A as elements with the property A not being element of A . He communicated to Cantor, to determine, which of the statements is true: R € R or R not € R. You can verify easily that there is a contradiction in any case.

It took more than fifty years to fix this logical bug. In the modern foundation of sets there is a difference between sets and classes. A set M is well defined, if it is an element in some existing class K. We write M € K. Creating a new class you have to make sure, that each element is a subset of an existing well defined set.

These definitions are quite complicated to work with. We skip the details and list the rules, which have come out:

Suppose  A is a set of sets.

union (A) = {x: there is a€A and x€a}

 intersection(A)= {x: for each  a€A is x€A. }

A  \ B = {x € A and x not € B}.

With these definitions the set A must not be empty for intersections. If A is empty, it violets the rule that each set comes as member of a class.


If A is a set with a finite number of elements you can use the following notation also:

a1 union a2 union a3.
a1 intersection a2.

Now it is necessary to assure the existence of some interesting sets to apply the operations above.

We postulate the existence of one set A at least axiomatically. Philosphically the words of the French mathematician Descartes "Cogito, ergo sum." might be a suitable motivation for that.

If you have convinced yourself of the existence of such a set A, you may define the empty set along the lines of Russell by application of the logical statement x € A "and" x "not" € A.

We call the set S a subset of M if it can be proved that each element in S is also an element in M. Starting with an arbitrary set M, we can define the class P(M) of all subsets in M. As an exercise for the reader we leave the proof that P(M) is a set.

Using the axiomatic approach to set theory we redefine the integer numbers recursively by the following definition: n(0) = {}; n(i+1)=n(i) "united with" {n(i)}. For the reader we leave the proof of the following assertions: 1. All expressions n(i) form well defined sets. 2.The class N of of all those sets can be verified also to be a set.

Now everybody of us should have understood that starting with the empty set there is a constructive method to generate for each integer number its unique successor.

The reader may prove that there are infinitely many elements in the set N, which are in one to one correspondence to integer numbers.

R is an order relation on a set M, if the following conditions are held:

Reflexivity: x R x for each element of M.

Antisymmetry: If x R y "and" y R x are true then x "equals" y.

Transitivity: If x R y "and" y R z are true then x R z is true.

The reader may prove that the set of integer numbers carries such an order relation.

In an ordered set it is not necessary that two elements are comparable. As an example we look at the set of subsets.

The reader should show that the set of integer numbers N carries a so called total order, where every two elements can be put in relation to each other.

We are interested in the existence of maximal and minimal elements.  Zorn's lemma gives the following  answer: If each  totally ordered subset has an upper bound , then there at least one maximal element (and vice versa lower bound for minimal elements). Zorn's lemma  is logically equivalent to the principle of choice, which makes sure that one can pick some elements out of any well defined set.

We define the product A x B  as set of all pairs (a,b) where a €  A "and" b € B and look at a  asubset  R of a product set. We write  a R b to express that (a,b) € R.

The  relation "=" is defined within a set G by the following subset {(x,x) € G x G}.

Generalizing this we come to a very important type of relation, called equivalence  relations. They have the following properties:

Reflexivity:  x ~ x is true for all x G.
Symmetry: if  x ~ y => y ~ x.
Transitivity: (x ~ y  "and" y ~ z)  => x ~ z.

We leave it to the reader to prove that "=" fullfills the properties of an equivalence relation.  Having  R we define for each x € G the set {y € G: y R x}. These sets are called equivalence classes with respect to  R.

We say that the set of equivalence classes of G with respect to R is G modulo R and use the following short notation G/R.

For a fixed divisor D  we call C " is rest of" A , if there is an integer B, such that A = B * D + C. 

The concept of maps is a very important. A map f from a set A to a set B is defined as relation f, that means to be a subset of A x B, which fullfills the following two conditions:

Well definition: For each x € A there is y € B with x f y

Uniqueness: If (a f b "and" a f c) => b=c .

There are some more optional properties of maps:

Injectivity: (x f y "and" z f y) => x=z.

Surjectivity: If for each y € B there is x € A "and" x f y.

Bijectivity: If f is injective and surjective.

Having a bijective map, we define the inverse map g from B into A by y g x <=> x f y .

Instead of the relational notation x f y we use the following form to write down a map alternatively: y = f(x).

The reader may look at the maps y = x² and y = x³ on the set of integers and decide, in which case there is an inverse.

It makes sense to measure the content of sets A by mapping it bijectively onto a subset of the integer numbers without gaps starting with the number one.

i) The reader may show that if there is a set G with finite measure n, then P(G), which is the set of all subsets, has 2 to the nth power many elements.

ii) All infinite subsets of the natural numbers are equivalent with respect to this measure.

As an exercsise for our readers we recommend to show that for a given  divisor B the relation "is rest of " is an equivalence relation in the set of integers.

In the text below we will define a bunch of mathematical objects. The following concept of categories allows a very efficient and general view onto the different classes of objects.

A category Cat consists of objects featuring certain common structures. A map from the object  A  to  B is called a morphism, if it preserves the structure.

We have to talk about classes Obj(Cat) with respect to objects, while the morphisms Morph(A,B) of a pair of objects form a set.

For objects  A,B, C and D of Obj(Cat), the following rules have to be met:

i) If f€Morph(A,B) and g€Morph(B,C), there is a map k € Morph(A,C) and k= f°g.

ii) Additionally, if h€Morph(C,D), then (f°g)°h = f°(g°h).

iii) There is a map n in Morph (B,B), such that f ° n = f and n°g = g.

The reader may verify that the rules are satisfied with the category of set, which will be denoted by Cat (set).

A set G carries a group structure, if there is a relation ° on G, such that the following properties are held:

Assoziativity: (g°h)°k = g°(h°k) for any three elements g,h and k.

Neutral element: There is a element e in G with e°g =g for each element g.

Inverse element: For each element g there is an element h such that g°h = e.

Commutativity: g°h = h°g is an optional property of a group.

We leave it to the reader to show that set of the bijective maps from A to B form a group.

Hint: Define the relation ° by using chaining of maps.

The reader may verify that the category Cat(group) is well defined.

If we have a relation, which meets only the law of associativity, the object is called a half group.

If there is an alphabet  A given as a finite set of symbols, the free halfgroup generated by these symbols by combining them to words will be denoted by HG(A).

My readers may prove that there are maps fn, such that the maps

fn:  HG ({1,...,n} )  -> N

are bijective.

Sometimes it is convienent to define an equivalence relation on  the the set of words of a halfgroup allowing to cancel a well defined subset.

We need to enhance the set N of natural numbers to implement groups structures with respect to addition and multiplication.

As far as addition is concerned we double the positive natural numbers by reflecting them with respect to zero. The resulting set is symbolized by the gothic letter Z.

The reader may verify that Z forms a commutative group.

We introduce the concept of rational numbers, which we symbolize by the letter Q, to make sure that there is an inverse operation for multiplication with the exception of zero. 


Q is defined to be the set of equivalence classes of Z x( Z - {0} ) using the following equivalence relation:

(a,b) ~ (c,d) <=> ad = bc.

The reader may verify that Q carries commutative group structures with respect to addition and multiplication.

Hint: Use the following definitions: (a,b) +(c,d)= (ad+bc, bd) and (a,b) *(c,d) = (ac,bd).

A set K is called a field, if there are commutative group structures for adding and multiplying respectively and the following distributive rule is valid for any three elements a,b and c:

Distributive rule a*(b + c) = a * b + a*c.

But are the rational numbers really perfect for arbitrary computions?

They are  not, because of the existence of numbers, which are not rational. If we look at a square with a side length of  a unit, then the diagonal's length is equal to the root of  two, which is not a fully computable number.

The formula of Hurwitz measures the exactness of approximation  by rational numbers. Before we are discussing this problem more in detail, we will  set up a grammar containing the rules of computing within a field:

expression = term [rterm];

rterm =( "+" | "-") term [rterm];

term = factor [rfactor];

rfaktor = ("*"|"/" ) factor [rfaktor];

factor = number | "(" expression ")"

Next we leave the domain of pure mathematics and analyze some important strategies to invest money as an application.

As long as money is good enough to buy all the necessities for living, travelling and entertaining, you will find people willing to work in order to earn money. At the same time selling new trend setting products and making cash cows in business even more profitable are well known ways to acquire fresh money.
 
If you have borrowed K units of money from a bank at an interest rate of r % and have agreed upon an annual payback plan with a constant rate R, the number of payments is an important figure in order to evaluate the cost the loan.

The growth of the capital after n years will be:

K (1+r) to the nth power ,

while the sum of the payback rates amounts to :

R ((1+r) to the nth power -1) / r .

We set up an equation with the expressions above at each side and get:

 n = log (R/(R - Kr)/log (1+r).

In order to keep the repayment periods short as possible you have to make sure that the redemption installment has been set to the maximum you can afford.

The following spreadsheet can be used to evaluate the cost of a loan:

Cost of a Loan Version 1.0

 

Exp.zip

The zip file above contains some pieces of a math workbench under construction.You will find the files lnR.ass and expR.ass with. W-assembler code to compute the natural logarithm and its inverse - which is the exponential funcion - up to 60 decimal digits.

Type in a Linux terminal window ./ LaderC lnR.ass in order to compute the natural logarithm or ./LaderC expR.ass to compute the inverse.
 

For a loan the number payments is n+1. From the view of the bank. K units of money have been lent out and n rates are to be paid back. Some banks allow tayloring of payments according to their customers needs.

In general we want to deterrnine the special rate of interest, which makes a number of positive or negative money movements balanced. It is called "Return on Investment".The  ROI is related to a fixed time interval, which might be a day or a month for example.

It can be computed from the x-values of a polynomial function, where the y-values are zero.

We return to the presentation of the mathematical theory, in especially we show how to compute the function log (x).

 A  normal form representing a rational number is a pair of numerator and denominator, where the greatest common divisor of the numbers  is equal to one. For computing  we need a procedure to determine the greatest  common divisor of two arbitrary integers. My readers may find out the details about the Euclidean algorithm, which is known since more 2000 years.

For computing purposes the presentation of rationals by dual  numbers is standard.  The information contained in  a dual number can be measured by the number of basic information units (bit).

Information of one bit will be generated by answering a single  question using the words „yes“ or „no“ only.

Now, we sketch a proof that the set of real numbers is not countable and  start assuming that a  function f  from the set of integer numbers into the set of real numbers exists, which is injective and surjective. Under these circumstances we  construct a real number w, which is not element of the image domain of the function f:   The i-th binary digit of w shall be set as the opposite of the it-h binary digit of f(i).

For my readers I leave it as an exercise to show that w is not in the  image domain of f. So we get  a contradiction to surjectivity, and the assumption must have been wrong.

We try to understand, in which way rational numbers are embedded within the set of reals. For this purpose we need to study functions  d(x,y) from X to R, which  measure  distances of two elements in a set X by real numbers, where X is an arbitrary set.

We call such a function a metric on X, if the following conditions are held.

i) d(x,x) = 0 for all x€X;
ii) d(x,y)=d(y,x) for all x,y€X;
iii)  d(x,z) <= d(x,y) + d(y,z) for all x,yz€X.


The third condition is referenced in the literature as triangle inequality.

For each set X you can define the following trivial metrics:

d1(x,y) = 0 if x=y and d1(x,y )=1 if x and y are different elements of X.

d2(x,y)=0 for all elements x and y of X.

Using a metric we can define neighborhoods containing some elements, which are situated nearby. We say that  a  subset U of X is open, if for x€U there is a open sphere  S(m,r )={x€X: d(x,m)<r}, which is contained  in U.

The complement of an open set  U with respect to X is X \ U. It is  a closed set by definition.

My readers may show that the property „open“ is compatible with union and intersection of sets and answer the question, which of the following options define a metric for pairs of real numbers (x,y):

Euclid((x1,y1),(x2,y2))= square root((x1-x2)²+(y1-y2)²),
max((x1,y1),(x2,y2))= maximum(|x1-x2|,|y1-y2|),
min((x1,y1),(x2,y2)) = minimum(|x1-x2|,|y1-y2|).

We are going to define a bunch of metrics for integer numbers n and m: Let p be a prime number. Then the distance between n and m will be measured by dp(n,m)=1/(2 to the k-th power), where k is the greatest common exponent of p in m and n.

There are some interesting applications:

1) Langenscheidt's "Großes Studienwörterbuch Englisch" contains about 315000 entries  and idioms. Having a list of prime numbers, the entries of the dictionary and the primes can be set  in one to one correspondence.  The metric above is suited to measure the distance between two documents n and m in a library with respect to a given word represented by p.

2) For the Latin alphabet containing 26 letters, we can measure the distance between two words or documents with respect to a certain letter.

Generalizing the concept of open sets we call a collection  T of open sets a topology, if

i) {}€T and X€T.
ii) For each set A, which is a subset of T, the union(A) € T,
iii) For each finite, nonempty set A, which is a subset of T, the intersection (A) € T

Remark: While unions according to ii) may extent to an arbitrary cardinal number of open sets, intersections according to iii) must be limited to a finite number. If the number of unions shall be limited too, it is convenient to use the concept of a base of a topology:

A base B of a topology T is a subset of open neighborhoods, which generates T only by application of the second axiom.

A base is called countable, if  there is a bijective map f from the natural numbers N to B.

It is possible to define a topology on a set X, if a metric is specified.

Conversely, we are interested to learn about the properties of those topologies, which are generated by a metric.

The three mathematicians Bing, Smirnov and Nagata gave the following answer:

The topologies of metric spaces are regular and they are generated by a countable base. Regularity says that for each closed set C and each point p, which is not in C, there are disjunct neighborhoods U1 containing C and p€U2.

If T1 and T2 are topologies on  X an each open set in T1 is also in T2 , we say that T2 is finer than T1 and write T1<T2.

The reader may show that this definition meets the axioms of an order relation.

We have learned that real numbers include the rational numbers as fractions of  integer numbers. Usually they are communicated as decimals having powers of basis ten as denominators  It depends on the basis, whether a communicated rational number can be expressed by fintely many symbols.

The reader may prove that solutions of the equations  x² = -1 or x²=2 cannot be presented as such fractions.

In order to find solutions we have to  complete the set of Q of rational numbers using two different methods. 

Next we will combine algebra and topology. This means that we are looking at the consequences of the interaction of the principles of algebraic abstraction and topological neighborhood.  

It should be remarked that you may define different topologies on a certain set.

A map a from the set of integers N to a topologic space  X is called a sequence. We say that a sequence a has a limit point p, if for each open set V in the topology of X containing p , all images  a(n) are in V with a finite number of exceptions. If p is unique, we say that a  converges to p.

Remark:  Even if a sequence has an infinite number of members, you can decide about convergency presenting an open set, which contains all the members with a finite list of exceptions.

The reader may figure out the limit point in the set of real numbers for a(n) = (1+5n)/(2n+33), b(n) = ((n+1)/n) to the n-th power and the sum of all coefficients c(n)=1/(q to the n-th power).

Hint: First find an algebraic expression for finite sums. 

We need the following definitions in order to demonstrate, in which way the rational numbers are embeded in the set of real numbers.

 Let A be a topological space and H a subset of A.  We say that H is dense in A, if   each neighborhood in A  contains a point of H.

Let Q be the set of rational numbers and S a sequence in Q. We say that a sequence S is of Cauchy type, if for each positve d€Q there is an integer n, such that |S(i)-S(j)|< d for all integers i and j , which are greater than n.

The reader may define a suited equivalence relation on the set of Cauchy sequences in Q and construct a bijektion between the set of equivalence classes and real numbers.

We define the field of complex numbers on the set of pairs of real numbers.

Addition: (a,b) + (c,d) = (a+c, b+d)

Multiplication: (a,b) * (c,d) = (ac-bd,ad+bc)

The components  of a complex number are called the real part and the imaginary part. We denote the imaginary unit (0,1) by the acrynom i.

Having a field there are neutral elements and we can compute the inverse of an element in both cases.

The readers may identífy the neutral elements and prove that in the first case the inverse of z=(a,b) is (-a,-b) and in the second case  (a/|z|),-b/|z|), where |z| is called the absolute value of z. It is the length of the vector pointing  from (0,0) to the complex number (a,b) according to Pythagoras.

Let f be a map from A to B with topologies TopA and TopB. We say that f is continuous at a point  a€ A, if for each neighborhood N of the image f(a) there is a neighborhood M of a such that f( M) is a subset of N.

We will refer to the set of continuous maps from A to B by writing C(A,B).

In topological spaces with a metric d continuity of a map f in a point x0 can be figured out in terms of real numbers. For this purpose for each positve number epsilon it has to be proved that there is a positive number delta(x0) meeting the following critereon:  d(x1,x0)< delta(x0) implies d(f(x1),f(x0))< epsilon.

The choice of the parameter delta may be dependend on epsilon and the base point x. If  delta can be chosen independendly of the base point, we refer to the term absolute continuity.

Starting with a map f: A -> B, where TopA is a well defined topology on A, B can be topologized in a way that f is continuous. The maximal topology of that kind is called the final topology via f.

Conversely if a topology TopB  on B is given, the minimal topology on A, which makes f continuous, is called the initial topology on A via f.

My readers may extent the definitions of initial and final topologies, if more than one map is involved.

We look at the category of topological spaces together continuous maps as morphisms, and define a sum and a product of a disjunctive pair of objects A and B:

A + B = (A union B) equipped with the final topology with respect of the embeddings.

A x B = ( A x B ) equipped witrh the initial topolgogy with respect to the projections.

Now we chose one special point in each space and mark it. We define the following operations:

(A,a) v (B,b) = (A + B) /( a~b)
(A,a) x (B,b) = (A x B, (a,b))
(A,a) ^ (B,b) = (A,a)  x (B,b) / ((A,a)  v  (B,b))

An algebraic module M over a ring R  is a  commutative group structure on M together with a multiplication of elements of A and M according to the axioms of distributivity (r1 + r2) m =  r1 m + r2 m and r ( m1 + m2) = r m1 + r m2. For the neutral element n of the ring with respect to multiplication the product n m  equals m.

A module is a vector space, if each element in the ring R has an inverse. This assertion means that R is a field.

A finite subset  {m1,..,mk} of M is called a set of generators, if each element has a representation  of the form 1*m1 + ... + ak*mk.

For a vector space  V a minimal set of  such generators is called a basis. The number of elements in a basis is the dimension of V. We write k=dim(V).

The reader may show that the field of real numbers R can be looked at as a vector space over itself  by the following definitions for adding and multiplying vectors:

The sum of two vectors v1 and v2 shall be set to cubic root(v1³ +v2³) and the product of a real number r times a vector v shall be set to the cubic root (r) times v .

The reader may show that the complex numbers are a vector space of dimension 2 over the field of real numbers.

The set A(V) is called an affine space if for each pair of elements (p1,p2) there is a vector v(p1,p2) in V such that v(p1,p2)+v(p2,p3)=v(p1,p3).

The set P(V) is called  an projective space if each subspace of Dimension one is element of P(V).

The reader may look at the projective space of a two dimensional complex vector space. Each subspace of complex dimension one may be characterized by a complex number a.

A map P from the set of integers to an algebraic ring R is called a power series. We say that a power series is a polynomial, if there are only a finite number of images different from zero. If there is only a single integer m, which is not equal to zero, we refer to it as  a monomial.

The reader may prove that polynomials and power series over a ring R form new rings. We refer to these rings by writing R[X] for polynomials and R[[X]] for power series.

We look at the Latin alphabet, where the letters and the symbols " " and "." are written in circular form, such that the letters are ordered lexicographically. There are 26 symbols altogether. We construct a map from words to polynmials like follows: "." -> 0," " ->  1, A  ->  X, B  ->  X²,  C  ->  X³ ...

In order construct the image of the map for an explictly given word, we give a listing of the procedure:

toPolynomial () {

var p[10]; number;
const space=32;

begin
   i:=1;
   read letter;
   while not letter=space and i<10 
   begin
      p[i]:=letter-64;
      i:=i+1;
     read letter
   end;  
end;}
  




 

We interpret the images of polynomials over an arbitrary ring R as coefficients and get maps from R to R of the following form: y =p(0) + p(1) x + ... p(n) x to the n-th power.

If we try to generalize the method above from the set of polynomials to all power series over the field of real or complex numbers respectively, we have to sum an inifinite number of monomials in order to figure out the y-value.

In order to handle infinite sums,  we need a topology of course.

As a task to do  mathematical calculations without electronic assistance the reader may  approximate the following y-values: 

the exponential series y=exp(x)= 1+x/1! + x²/2! +x³/3! + ... using x-values 1, 1.01 and 5,

and the logarithmic series y=log(x)= 1- 1/2 x² + 1/3 x³ ... using the x values 1/2  and 2. 

The reader may determine the set C  of x-values, which render finite y-values. C is known as the domain of convergency.

Based on the complex exponential function we can define the  following real functions:

cos(x)=(exp(ix) + exp(-ix))/2

sin(x)= (exp(ix) -  exp(-ix))/2i

cosh(x) =(exp(x) + exp(-x))/2

sinh(x)=(exp(x) - exp(-x))/2

For the complex element (0,1) we have written the letter i.

We are looking for an algorithm to calculate the coefficients for power series, if a map f is given.

Our first approach does only work, if f is a smooth map.

The following concept was founded by Leibniz and Newton. (The names appear in alphabetical sequence.)

A map f is called smooth at a point x, if you can approximate it by a linear map L in a certain neighborhood of f(x).  A map is linear, if L(a+b) = L(a)+L(b) and L(kr)=k L(r) are true for all a,b,k,r € R.

The stereographic projection of the two dimensional sphere onto the set of complex numbers is the focus next.

Our ensemble is build by  the complex plane with a sphere of radius one touching at zero. For each point  p of the sphere we draw a line from the north pole  to p and take the point z, where the line pierces  the plane as image of p.

There is no image defined for the north pole itself.

The distance  d of two points p and q on the sphere can be  transferred to the plane using the stereographic projection.

The reader may figure out the details in terms of algebra and proof that the transfered metric from the sphere is bounded, while the Euclidean metric is not.

The reader should verify that the north pole is the limit point of the sequence of integers in the complex plain, which is stereographically projected to the sphere.
If we add a new point called infinity to the complex plane, which corresponds to the north pole, we have to extent the algebraic rules for complex numbers like follows:

z + infinity = infinity
z -  infinity = infinity
z * infinity =infinity
z / infinity = 0
z / 0     = inifinity         if z is not equal to zero
            = undefined      if z = 0

Now we will look at the complex functions of the form: (a z + b)/(c z + d) with ad-bc = 1.

It is left to the reader to convince herself or himself that the functions above can be extentend to the sphere using the stated rules for calculation with infinity.

The reader may prove that the functions defined above form a group by composing them like follows: f1 ° f2 = f1 (f2(z)).

We look at the equivalence classes of the sphere, the complex numbers or the open unit circle with respect to the following relation R: z R w is true, if a function w =(az+b)(/(cz+d) exists, such that ad-bc=1, and w=f(z).

A subset K of these functions may form a subgroup, such that you can find a neighborhood Uz for each point z, which has a not empty intersection with a finite number images of other point's neighborhoods mapped by group members to z only.

Subgroups of this kind are called discontinuous.

They are called Kleinian groups, Euclidean groups or Fuchsian groups according to their domain of operation (sphere, Complex numbers or unit circle).

We want to make sure that a given complex domain does not contain any gaps or holes. Then each closed curve can be deformed continuously to a single point.

Riemann proved, that each domain of the described kind can be mapped either to the sphere, the complex numbers or the unit circle using a map, which preserves angles. Such a map is called a conformal map.

We have studied the basics in the category of topological spaces. We have stated the necessary and sufficient properties that such a space is generated by a metric. We have learned that metrics  allows to measure the distance between two points by a real number.

We can use a metric to define a measure for intervals in real vector spaces of finite dimension: The measure µ for one dimensionial intervalls from a to b is defined by µ(a,b)= d(a,b). Each n dimensional interval I is the product set of one dimensional intervals:  I= I1 x ... x In. We set µ(I) = d1(I1) * ... *dn(In).

But we would like to study more generalized measures, which we need in applied mathematics with the following properties:

i) µ(A) >= 0 (positivity)

ii) µ(A) <= µ(B) if A is a subset of B (monotony)

iii) µ(A union B) = µ(A) + µ(B) if A and B have no common points (additivity)

iv) For each measurable set A there is for each d an open neighborhood B such that µ(B) -µ(A) < d (regularity) 

While in topology we have studied a generalized theory of sets, having added a concept of distance, which allows to have a look upon local strucutures,  now we want to develop a theory of measuring the content of well defined subsets in a space X.

We will learn that there are some subsets of a space X, which are not measurable.

We call a collection M of measurable subsets of X a sigma algebra, if the following conditions are met:

i) The empty set is measurable.
ii) If A € M, the complement of A relative to X is measurable.
iii) If A € M and B € M, it folllows that the intersection of A and B is measurable.
iv) For a countable subcollection L the union of members is measurable.

The real vector spaces carry a natural sigma algebra B, which is generated by the intervalls. The members are called Borel sets.

The reader should verify at each countalbe subset of X is of measure zero.

Having two sets A and B, differing by a set N with µ(N)=0 only, we say that A and B are equivalent with respect to the measure µ.

We need some definitions yet:

A map f: (X,M,µ1) ->Y (Y,K,µ2) is called measurable, if for each measurable set B in Y there is a measurable set A in X, and f(A)=B.

It is left to the reader to show that a given sigma algebra on Y gives rise to a sigma algebra on X.

A primitive function from a measurable set A to the real numbers maps the elements of A to one, and all other points go to zero.

Next we will show in which way a measurable function f from X to the reals can be approximated by primitive functions.

The triple (X,M,µ) is called probability space, if  M is a sigma algebra on X and µ:  M into the closed real interval [0,1] is a measure with µ(X)=1.

A measurable function f from a probablity space to the real line is called random variable.

Traditionally probability was based on the quotient S/N of the sucessfull versus total number of resulting events. But this definition does not apply, if the number of outcoming events is not finite.

IN the following text we will use the capital letter P as acrynm for probability instead of µ.

We will look at some very basic schemes of acting and construct the appropriate probablity spaces:


A coin is thrown:

 X={nummber, image}; P(number)=P(image)=1/2.

A number wheel is turned:

i) each number occurs only once:
 X={1,... n}; P(i)=1/n

ii) the number i occurs h(i) times:
   X={1,... n}; P(i)=h(i)/n

In the U. S. there was a quiz show on TV, where the candidate with the most points could win an automobile. The car was in one of three numbered garages.The person was asked to  to pick a number.  Then the quiz master opened an empty garage different from the figure chosen by the candidate.

Finally the person had the option to change his mind and pick a different number.

The reader may  construct the appropriate probability space.

12/26/2016

W. Reiwer