微积分的历史演变与现状

 微积分的历史演变与现状

         站在现代数学的高度观察微积分学的历史演变与现状，实事求是，还原历史原来的面貌，具有极大的现实意义。

J.keisler在其巨著《基础微积分》后记中做到了这一点。

袁萌   10月7日

附：基础微积分

EPILOGUE（后记）

How does theinfinitesimal calculus as developed in this book relate to the traditional (orε, δ ) calculus? To get the proper perspective we shall sketch the history ofthe calculus. 

Many problemsinvolving slopes, areas and volumes, which we would today call calculusproblems, were solved by the ancient Greek mathematicians. The greatest of themwas Archimedes( 287-212 B.C.) . Archimedes anticipated both the infinitesimaland the ε, δ approach to calculus. He sometimes discovered his results byreasoning with infinitesimals, but always published his proofs using the“method of exhaustion,” which is similar to the ε, δ approach.

Calculus problemsbecame important in the early 1600’s with the development of physics andastronomy. The basic rules for differentiation and integration were discoveredin that period by informal reasoning with infinitesimals. Kepler, Galileo,Fermat, and Barrow were among the contributors.

In the 1660’s and1670’s Sir Isaac Newton and Gottfried Wilhelm Leibniz independently “invented”the calculus. They took the major step of recognizing the importance of acollection of isolated results and organizing them into a whole.

Newton, atdifferent times, described the derivative of γ (which he called the “fluxion”of γ) in three different ways, roughly

(1) The ratio ofan infinitesimal change in y to an infinitesimal change in x ( Theinfinitesimal method.)

 

(2) The limit ofthe ratio of the change in y to the change in x, Δy/Δx, as Δx approaches zero.(The limit method.)

(3) The velocityof y where x denotes time. (The velocity method.)

In his laterwritings Newton sought to avoid infinitesimals and emphasized the methods (2)and (3).

Leibniz ratherconsistently favored the infinitesimal method but believed (correctly) that thesame results could be obtained using only real numbers. He regarded theinfinitesimals as “ideal” numbers like the imaginary numbers. To justify themhe proposed his law of continuity: “In any supposed transition, ending in anyterminus, it is permissible to institute a general reasoning, in which theterminus may also be included.” This “law” is far too imprecise by presentstandards. But it was a remarkable forerunner of the Transfer Principle onwhich modern infinitesimal calculus is based. Leibniz was on the right track,but 300 years too soon!

The notationdeveloped by Leibniz is still in general use today, even though it was meant tosuggest the infinitesimal method: dy /dx for the derivative (to suggest aninfinitesimal change in y divided by an

infinitesimal change in x ), and ∫baƒ(x )dx for theintegral (to suggest the sum of infinitely many infinitesimal quantities ƒ(x ) dx ).

All threeapproaches had serious inconsistencies which were criticized most effectivelyby Bishop Berkeley in 1734. However, a precise treatment of the calculus was beyondthe state of the art at the time, and the three intuitive descriptions(1)-(3)of the derivative competed with each other for the next two hundred years.Until sometime after 1820, the infinitesimal method (1) of Leibniz was dominanton the European continent, because of its intuitive appeal and the convenienceof the Leibniz notation. In England the velocity method (3) predominated; italso has intuitive appeal but cannot be made rigorous.

In 1821 A. L.Cauchy published a forerunner of the modern treatment of the calculus based onthe limit method (2). He defined the integral as well as the derivative interms of limits, namely

 

∫baƒ(χ)dχ=lim Σƒ(χ)Δχ

 

He still usedinfinitesimals, regarding them as variables which approach zero. From that timeon, the limit method gradually became the dominant approach to calculus, whileinfinitesimals and appeals to velocity survived only as a manner of speaking.There were two important points which still had to be cleared up in Cauchy’swork, however. First, Cauchy’s definition of limit was not sufficiently clear;it still relied on the intuitive use of infinitesimals. Second, a precisedefinition of the real number system was not yet available. Such a definitionrequired a better understanding of the concepts of set and function which werethen evolving.

A completelyrigorous treatment of the calculus was finally formulated by Karl Weierstrassin the 1870’s. He introduced the ε, δ condition as the definition of limit. Atabout the same time a number of mathematicians, including Weierstrass,succeeded in constructing the real number system from the positive integers.The problem of constructing the real number system also led to development ofset theory by Georg Cantor in the 1870’s. Weierstrass’ approach has become thetraditional or “standard” treatment of calculus as it is usually presentedtoday. It begins with the ε, δ condition as the definition of limit and goes onto develop the calculus entirely in terms of the real number system(with nomention of infinitesimals). However, even when calculus is presented in thestandard way, it is customary to argue informally in terms of infinitesimals,and to use the Leibniz notation which suggests infinitesimals.

From the time ofWeierstrass until very recently, it appeared that the limit method(2) hadfinally won out and the history of elementary calculus was closed. But in 1934,Thoralf Skolem constructed what we here call the hyperintegers and proved thatthe analogue of the Transfer Principle holds for them. Skolem’s construction(now called the ultraproduct construction) was later extended to a wide classof structures, including the construction of the hyperreal numbers from thereal numbers.

 

'See Kline, p.385. Boyer, p. 217.

 

The name“hyperreal” was first used by E. Hewitt in a paper in 1948. The hyperrealnumbers were known for over a decade before they were applied to the calculus.

Finally in 1961Abraham Robinson discovered that the hyperreal numbers could be used to give arigorous treatment of the calculus with infinitesimals. The presentation of thecalculus which was given in this book is based on Robinson’s treatment (butmodified to make it suitable for a first course).

Robinson’scalculus is in the spirit of Leibniz’ old method of infinitesimals. There aremajor differences in detail. For instance, Leibniz defined the derivative asthe ratio Δy/Δx where Δx is infinitesimal, while Robinson defines thederivative as the standard part of the ratio Δy/Δx where Δx is infinitesimal.This is how Robinson avoids the inconsistencies in the old infinitesimalapproach. Also, Leibniz’ vague law of continuity is replaced by the preciselyformulated Transfer Principle.

The reasonRobinson’s work was not done sooner is that the Transfer Principle for the hyperrealnumbers is a type of axiom that was not familiar in mathematics until recently.It arose in the subject of model theory, which studies the relationship betweenaxioms and mathematical structures. The pioneering developments in model theorywere not made until the 1930’s, by GÖdel, Malcev, Skolem, and Tarski; and thesubject hardly existed until the 1950’s.

Looking back wesee that the method of infinitesimals was generally preferred over the methodof limits for over 150 years after Newton and Leibniz invented the calculus,because infinitesimals have greater intuitive appeal. But the method of limitswas finally adopted around 1870 because it was the first mathematically precisetreatment of the calculus. Now it is also possible to use infinitesimals in amathematically precise way. Infinitesimals in Robinson’s sense have beenapplied not only to the calculus but to the much broader subject of analysis.They have led to new results and problems in mathematical research. SinceSkolem’s infinite hyperintegers are usually called nonstandard integers.Robinson called the new subject “nonstandard analysis.” (he called the realnumbers “standard” and the other hyperreal numbers “nonstandard.” This is theorigin of the name “standard part.”)

The startingpoint for this course was a pair of intuitive pictures of the real andhyperreal number systems. These intuitive pictures are really only roughsketches that are not completely trustworthy. In order to be sure that theresults are correct, the calculus must be based on mathematically precisedescriptions of these number systems, which fill in the gaps in the intuitivepictures. There are two ways to do this. The quickest way is to list themathematical properties of the real and hyperreal numbers. These properties areto be accepted as basic and are called axioms. The second way of mathematicallydescribing the real and hyperreal numbers is to start with the positiveintegers and, step by step, construct the integers, the rational numbers, the realnumbers, and the hyperreal numbers. This second method is better because itshows that there really is a structure with the desired properties. At the endof this epilogue we shall briefly outline the construction of the real andhyperreal numbers and give some examples of infinitesimals.

We now turn tothe first way of mathematically describing the real and hyperreal numbers. Weshall list two groups of axioms in the epilogue, one for the real numbers andone for the hyperreal numbers. The axioms for the hyperreal numbers will justbe more careful statements of the Extension Principle and Transfer Principle ofChapter 1. The axioms for the real numbers come in three sets: the AlgebraicAxioms, the Order Axioms, and the Completeness Axiom. All the familiar factsabout the real numbers can be proved using only these axioms.

1.ALGEBRAICAXIOMS FOR THE REAL NUMBERS

   A    Closure laws 0 and 1 are realnumbers. If a and b are real numbers, then so are a + b, ab and -a. If a is areal number and a ≠ 0, then 1/a is a realnumber.

 

B    Commutative laws     a+b =b+a  ab = ba

C    Associative laws     a+(b+c) = (a+b) +c     a(bc) = (ab) c.

D    Identity laws        0+a =a               1· a = a .

E    Inverse laws         a +(-a)=0     if a ≠0,     a. 1                                                     

F    Distributivelaw        a·(b + c) = ab + ac

 

DEFINITION

         The positive integers are the real numbers 1,2 = 1+1, 3 = 1+1+1 ,

          4= 1+1+1+1, and so on.

║.  ORDER AXIOMS FORREAL NUMBERS

A    0<1.

B    Transitive law  if a< b and b< c.

C    Trichotomy law Exactly one of the relations a

D    Sum law  If a< b , then a+c < b+c.

E    product law  If a < bc.

F    Root axiom   For every real number a>0 and every positive integer n, there is a real number b>0 such that _____=a

 

Ⅲ.  COMPLETENESS  AXIOM

 

    Let A be a set of real numbers such that wheneverx and y are in A, any real number between x and y is in A. Then A is aninterval.

 

THEOREM     

An increasing sequence <</font>___ > either converges ordiverges to ∞.

 

PROOF  let T be the set of all real number x such that x ≤____ for some n.T is obviously nonempty.

 

Case 1  T is the whole real line. If H is infinite we have x ≤____ for all realnumbers x. So __ is positive infinite and <</font>__ > diverges to ∞

.

Case 2   T is not the whole real line. By the CompletenessAxiom, T is an interval (－∞, b] or (－∞, b). For eachreal x < b, we have

 

 

 

     for some n. It follows that for infiniteH, __ ≤ b and __≈b. Therefore<</font>__> converges to b.

 

We now take up the second group of axioms, which give the properties ofthe hyperreal numbers. There will be two axioms, called the Extension Axiom andthe Transfer Axiom, which correspond to the Extension Principle and TransferPrinciple of Section 1.5. We first state the Extension Axiom.

 

1*. EXTENSION AXIOM

          (a ) Theset R of real numbers is a subset of the set R* of hyperreal numbers.

          (b) Thereis given a relation <* on R*, such that the order relation < on R is asubset of <*, <* is transitive (a <*b and b <* c implies a<* c),and <* satisfies the Trichotomy Law: for all a ,b in R*, exactly one ofa<* b, a = b, b<* a holds.

 

(c) There is a hyperreal number ε such that 0<* ε and ε< * r foreach positive real number r.

(d) For each real function f, there is given a hyperreal function f*with the same number of variables, called the natural extension of f.

 

Part (c ) of the Extension Axiom states that there is at least onepositive infinitesimal. Part (d) gives us the natural extension for each real function.The Transfer Axiom will say that this natural extension has the same propertiesas the original function.

 

Recall that the Transfer Principle of Section 1.5 made use of theintuitive idea of a real statement. Before we can state the Transfer Axiom, wemust give an exact mathematical explanation of the notion of a real statement.This will be done in several steps, first introducing the concepts of a realexpression and a formula.

 

We begin with the concept of a real expression, or term, built up fromvariables and real constants using real functions. Real expressions can bebuilt up as follows:

(1) A real constant standing alone is a real expression.

A variable standing alone is a real expression.

 If e is a real expression, and f is a real function of onevariable, then f(e) is a real expression. Similarly, if ____________ are realexpressions, and g is a real function of n variables, then g(_________) is areal expression.

 

Step(3) can be used repeatedly to build up longer expressions. Here aresome examples of real expressions, where x and y are variables:

 

 

 

 

By a formula, we mean a statement of one of the following kinds, whered and e are real expressions:

(1)An equation between two real expressions, d = e.

(2)An inequality between two real expressions, d< e, d≤e, d> e, d≥e, or d≠e.

 

（3）A statement ofthe form “ e is defined” or “e is undefined.”

 

Here are some examples of formulas:

 

                    x + y= 5,

 

                 f (x)=  1－x²

                          1+x²

                 g (x,y ) <</i> f (t),

                 f (x, x) is undefined.

 

If each variable in a formula is replaced by a real number, the formulawill be either true or false. Ordinarily, a formula will be true for somevalues of the variables and false for others. For example, the formula x + y =5will be true when (x, y)=(4,1) and false when (x,y) =(7, －2).

 

DEFINITION

 

A real statement is either a nonempty finite set of formulas T or acombination involving two nonempty finite sets of formulas S and T that statesthat “whenever every formula in S is true, every formula in T is true.”

 

We shall give several comments and examples to help make thisdefinition clear. Sometimes, instead of writing “whenever every formula in S istrue, every formula in Tis true” we use the shorter form “if S then T” for areal statements.Each of the Algebraic Axioms for the Real Numbers is a realstatement. The commutative laws, associative laws, identity laws, anddistributive laws are real statements.

For example, the commutative laws are the pair of formulas

 

                            a+ b = b+a,     ab=ba,

Which involve the two variables a and b. The closure laws may beexpressed as four real statements:

                               a +b is defined,

                              abis defined,

                               -ais defined,

                                If a≠0, then 1/ais defined.

 

The inverse laws consist of two more real statements. The TrichotomyLaw is part of the Extension Axiom, and all of the other Order Axioms for theReal Numbers are real statements. However, the Completeness Axioms is not areal statement, because it is not built up from equations and inequalitiesbetween terms.

 

A typical example of a real statement is the inequality for exponentsdiscussed in Section 8.1:

 

                 if a ≥ 0, and q≥ 1, then(a+1)q≥ aq+1

 

This statement is true for all real numbers aand q.

 

A formula can be given a meaning in the hyperreal number system as wellas in the real number system. Consider a formula with the two variables xand y.When x and yare replaced by particular real numbers, the formula in thehyperreal number system. To give the formula a meaning in the hyperreal numbersystem, we replace each real function by its natural extension and replace thereal order relation < by the hyperreal relation<*. When x and yarereplaced by hyperreal numbers, each real function fis replaced by its naturalextension f *, and the real order relation <</span>is replacedby<*, the formula will be either true or false in the hypereal numbersystem.

 

For example, the formula x + y=5 is true in the hyperreal number systemwhen(x,y)=(2－ε, 3 + ε),

 

but false when (x,y ) = (2+ε, 3+ε) , if εis nonzero.

 

We are now ready to state the Transfer Axiom.

 

║* .  TRANSFER AXIOM

 

Every real statement that holds for all real numbers holds for allhyperreal numbers.

 

It is possible to develop the whole calculus course as presented inthis book from these axioms for the real and hyperreal numbers. By the TransferAxiom, all the Algebraic Axioms for the Real Numbers also hold true for thehyperreal numbers. In other words, we can transfer every Algebraic Axiom forthe real numbers to the hyperreal numbers. We can also transfer every OrderAxiom for the real numbers to the hyperreal numbers. The Trichotomy Law is partof the Extension Axiom. Each of the other Order Axioms is a real statement andthus carries over to the hyperreal numbers by the Transfer Axiom. Thus we canmake computations with the hyperreal numbers in the same way as we do for thereal numbers.

 

There is one fact of basic importance that we state now as a theorem.

 

THEOREM  (Standard  Part  Principle)

 

 For every finite hyperreal number b, there is exactly one realnumber r that is infinitely close to b.

 

PROOF We first show that there cannot be more than one real numberinfinitely close to b.Suppose r and sare real numbers such that  r ≈ band s ≈ b. Then r ≈ s, and sincer  and s are real, r must be equal to s . Thus there is at most one realnumber infinitely close to b.

 

We now show that there is a real number infinitely close to b. Let Abethe set of all real numbers less than b. Then any real number between twoelements of Ais an element of A. By the Completeness Axiom for the realnumbers, Ais an interval. Since the hyperreal number bis finite. Amust be aninterval of the form(－∝, r ) or (－∝, r ]for somereal number r. Every real number s belongs to A,so s < b. Also, every realnumber  t > r  does not belong to A, so t ≥ b. This showsthat ris infinitely close to b.

It was pointedout earlier that the Completeness Axiom does not qualify as a real statement.For this reason, the Transfer Principle cannot be used to transfer theCompleteness Axiom to the hyperreal numbers. In fact, the Completeness Axiom isnot true for the hyperreal numbers. By a closed hyperreal interval, we mean aset of hyperreal numbers of the form [a, b], the set of all hyperreal numbersxfor which a≤ x ≤ b, where aandbare hyperreal constants. Open and half-open hyperreal intervals are defined ina similar way. When we say that the Completeness Axiom is not true for thehyperreal numbers, we mean that there actually are sets A of hyperreal numberssuch that:

(a) whenever xandyare in A, any hyperreal number between xand yis in A.

(b) A is not ahyperreal interval.

Here are twoquitefamiliar examples.

EXAMPLE1 The setAof all infinitesimals has property (a ) above but is not a hyperreal interval.It has property (a ) because any hyperreal number that is between twoinfinitesimals is itself infinitesimal. We show that Ais not a hyperrealinterval. Acannot be of the form [a, ∞] or (a,∞) because everyinfinitesimal is less than 1. Acannot be of the form [a, b] or (a, b], becauseif bis positive infinitesimal, then 2·b is a largerinfinitesimal. Acannot be of the form [a,b) or (a,b), because if bis positive andnot infinitesimal, then b/2 is less than b but still positive and notinfinitesimal.

The set B of allfinite hyperreal numbers is another example of a set that has property (a )above but is not an interval.

Here are someexamples that may help to illustrate the nature of the hyperreal number systemand the use of the Transfer Axiom.

 

EXAMPLE2 let f bethe real function given by the equation.

Its graph is theunit semicircle with center at the origin. The following two real statementshold for all real numbers x:

By the TransferAxiom, these real statements also hold for all hyperreal numbers x. Thereforethe natural extension  f* of f  is given by the same equation

The domain off*is the set of all hyperreal numbers between －1 and 1. Thehyperreal graph of f*, shown in Figure E.1, can be drawn on paper by drawingthe real graph of f(x)and training an infinitesimal microscope on certain keypoints.

EXAMPLE3 let fbethe identity function on the real numbers, f(x)=x. By the Transfer Axiom, theequation f(x)= x is true for all hyperreal x. Thus the natural extension f*offis defined, and f*(x )=x for all hyperreal x. Figure E.2 shows the hyperrealgraph of f*. Under a microscope, it has a 45°slope.

Here is anexample of a hyperreal function that is not the natural extension of a realfunction.

 

 

 

 

 

 

Figure E.1

 

 

 

Figure E.2

 

 

 

EXAMPLE4 One hyperreal function, which we have already studied in somedetail, is the standard part function st(x). For real numbers the standard partfunction has the same values as the identity function,

 

st(x) = x for all real x.

However, thehyperreal graph of st(x), shown in Figure E.3, is very different from thehyperreal graph of the identity function f*. The domain of the standard partfunction is the set of all finite numbers, while f* has domain R*. Thus forinfinite x, f*(x) = x, but st(x)is undefined. If xis finite but not real, f*(x)=x but st(x)≠ x. Under the microscope, aninfinitesimal piece of the graph of the standard part function is horizontal,while the identity function has a 45° slope.

The standard partfunction is not the natural extension of the identity function, and hence isnot the natural extension of any real function.

 

 

 

 

 

 

Figure E.3

The standard partfunction is the only hyperreal function used in this course except for naturalextensions of real functions.

We conclude witha few words about the construction of the real and the hyperreal numbers.Before Weierstrass, the rational numbers were on solid ground but the realnumberswere something new. Before one could safely use the axioms for the realnumbers, it had to be shown that the axioms did not lead to a contradiction.This was done by starting with the rational numbers and constructing astructure which satisfied all the axioms for the real numbers. Since anythingproved from the axioms is true in this structure, the axioms cannot lead to acontradiction.

The idea is toconstruct real numbers out of Cauchy sequences of rational numbers.

DEFINITION

A Cauchy Sequence is a sequence <</i>________> of numberssuch that for every real ε > 0 there is an integer ____ such that thenumbers

 

 

 

 

are all within ε of each other.

 

Two Cauchy sequences

 

 

 

of rational numbers are called Cauchy equivalent, in symbols<</span>_____> =<</span>____>, if the differencesequence

 

 

 

converges to zero. (Intuitively this means that the two sequences havethe same limit.)

 

PROPERTIES OF CAUCHY EQUIVALENCE

 

(1)If <</span>_______> =<</span>______> and< ______> = <</span>_______>

 then the sum sequences are equivalent.

 

 

(2)Under the same hypotheses, the product sequences are equivalent,

 

 

(3)if__ = __ for all but finitely many n, then

   

 

 

 

The set of real numbers is then defined as the set of all equivalenceclasses of Cauchy sequences of rational numbers. A rational number rcorresponds to the equivalence class of the constant sequence < r, r,r,…>. The sum of the equivalence class of <</span>___ > and theequivalence class of < ___ > is defined as the equivalence class of thesum sequence

 

 

 

The product is defined in a similar way. It can be shown that all theaxioms for the real numbers hold for this structure.

 

Today the real numbers are on solid ground and the hyperreal numbersare a new idea. Robinson used the ultraproduct construction of Skolem to showthat the axioms for the hyperreal numbers (for example, as used in this book)do not lead to a contradiction. The method is much like the construction of thereal numbers from the rationals. But this time the real number system is thestarting point. We construct hyperrealnumbers out of arbitrary (not justCauchy) sequences of real numbers.

 

By an ultraproduct equivalence we mean an equivalence relation= on theset of all sequences of real numbers which have the properties of Cauchyequivalence (1)-(3) and also

 

(4)If each __belongs to the set {0,1} the <</span>__> isequivalent to exactly one of the constant sequences <0,0,0,…> or<1,1,1,…>.

 

Given an ultraproduct equivalence relation, the set of hyperrealnumbers is defined as the set of all equivalence classes of sequences of realnumbers. A real number rcorresponds to the equivalence class of the constant sequence<</span>r, r,r…>. Sums and products are defined as for Cauchysequences. The natural extension f*of a real function f(x)is defined so thatthe image of the equivalence class of <</span>__> is theequivalence class of <</span>___>. It can be proved thatultraproduct equivalence relations exist, and that all the axioms for the realand hyperreal numbers hold for the structure defined in this way.

 

When hyperreal numbers are constructed as equivalence classes ofsequences of real numbers, we can give specific examples of infinite hyperrealnumbers. The equivalence class of

                         <1,2,3,……n,……>

is a positive infinite hyperreal number. The equivalence class of

                         <1,4,9……n²……>

is larger, and the equivalence class of

                         <1,2,4……__……>

is a still larger infinite hyperreal number.

We can also give examples of nonzero infinitesimals. The equivalenceclasses of

                    <1, 1/2, 1/3……1/n, ……>,

                    <1,1/4,1/9……n–², ……>,

and                 <1,1/2,1/4……___________, ……>,

are progressively smaller positive infinitesimals.

 

The mistake of Leibniz and his contemporaries was to identify all theinfinitesimals with zero. This leads to an immediate contradiction becausedy/dxbecomes 0/0. In the present treatment the equivalence classes of

                            

                         <1, 1/2, 1/3, ……, 1/n, ……>

and      <0, 0, ……0, ……>

are different hyperreal numbers. They are not equal but merely have thesame standard part, zero. This avoids the contradiction and once again makesinfinitesimals a mathematically sound method.

 

For more information about the ideas touched on in this epilogue, seethe instructor’s supplement, Foundations of Infinitesimal Calculus,which has aself-contained treatment of ultraproducts and the hyperreal numbers.

 

FOR FURTHER READING ON THE HISTORY OF THE CALCULUS SEE:

The History of the Calculus and its Conceptual Development: Carl c.Boyer, Dover, New York, 1959.

Mathematical Thought from Ancient to Modern Times; Morris Kline, OxfordUniv.

Press, New York, 1972.

No-standard Analysis: Abraham Robinson, North-Holland, Amsterdam,London, 1966.

 

FOR ADVANCED READING ON INFINITESIMAL ANALYSIS SEE NO-STANDARD ANALYSISBY ABRAHAM ROBINSON AND:

 

Lectures on Non-standard Analysis; M. Machover and J. Hirschfeld,Springer-Verlag, Berlin, Heidelberg, New York, 1969.

Victoria Symposium on Nonstandard Analysis; A. Hurd and p. Loeb,Springer-Verlag, Berlin, Heidelberg, New York, 1973.

Studies in Model Theory; M. Morley, Editor, Mathematical Association ofAmerica, Providence, 1973.

Applied Nonstandard Analysis: M. Davis, Wiley, New York, 1977.

Introduction to the Theory of Infinitesimals: K.D Stroyan and W.A. J.Luxemburg, Academic Press, New York and London, 1976.

Foundations of Infinitesimal Stochastic Analysis: K.D. Stroyan and J.M..Bayod, North-Holland Publ.Co., in press.