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1. The Task to Be Accomplished, revised 13/10/17 at 11:30 Page 1

Pretty Good Integration Theory

By

Dr. John William Poduska, Sr.

Abstract {205 words}

This is one of a series of “Pretty Good” papers. The subject is Integration Theory from the point-of-view of an Engineer or Scientist solving problems in the “Real World”. We do not abandon proofs, but we will often omit steps, or worse, give a loose description of a proof. Moreover, we choose approaches that set aside global generality in favor of simplicity and minimum deviation from a direct path to results. We want to know why integrals converge, and why the simple Riemann Integral is insufficient. On the other hand, the Lebesgue Integral is outrageously complex for the tasks we have at hand. We therefore dwell on the Henstock-Kurzweil (a.k.a. the Gauge, or extended Riemann) integral (which we will uniformly call the HK integral). We discuss integration in three ways: as anti-differentiation, as the area under a curve, and as the limit of a sum. We talk about Stieltjes form integrals in the context of Wiener Process Integrals as well as “Stochastic Differential Equations”. More than anything else, we seek simplicity and physical insight – goals that often conflict. The Thesis is this: Unless and until the complexity of measure theory be required, we should use and teach the HK integral for its vigor and simplicity.


1.1. A Little History and Background {493 words}, revised 13/10/17 at 11:30 Page 2

1. The Task to Be Accomplished

1.1. A Little History and Background {493 words}

Everyone reading this knows a lot about integration. For example, we know from High School

that the following integral “exists”:

3

2 xx dx Const

3 (1.1.1)

Moreover, we know the closed form integral of many functions, even Bessel functions and Mathieu

Functions and the like. There are huge books of integrals of functions to assist our failing memories.

Without exception (or at least none that I know of), we prove our integral equation by differentiating the

result to see if we get the original kernel integrand!

When we get into the Physics of the continuum, (sophomore physics) we encounter line

integrals, surface integrals, and volume integrals. When we get into Fourier and Laplace transforms

(sophomore electrical engineering) we see contour integrals with “principal values” etc. We even see

Riemann-Stieltjes integrals occasionally when we deal with step functions and impulse functions.

All of these are examples of either the “Newton Integral” (anti-differentiation) or the Riemann

Integral (a limit of a sum). Newton did his work on integration in the 1660’s (actually the differential

and integral calculus) and was involved in a nasty dispute with Leibniz. In 1854, Riemann presented his

integral as a well-defined limit of a sum.

For us (engineers and physicists) this is nearly the end of the story. We use the Newton and

Riemann Integrals for everything including numerical integration. We never have problems we attribute

to the method of integration.

BUT, our mathematician friends tell us there are serious problems with the Newton and Riemann

integrals. They tell us that these integrals lack the convergence properties required to use in Hilbert

Spaces (mostly orthogonal polynomial spaces). Worse, we cannot always move a derivative or sum or

limit under the integral sign. Thus:

b b

a a

b b

n na a

n n

b b

n na an n

d dF(z, x)dx ? F(z, x)dx

dz dz

F (z, x)dx ? F (z, x)dx

lim F (z, x)dx ? lim F (z, x)dx

(1.1.2)

This is serious. But, there is a “Hounds of Baskerville” element here – the dogs don’t seem to bark, we

never seem to get into trouble using the Newton and/or Riemann Integrals. Why?

Our mathematician friends say we should use the Lebesgue Integral (Henri Lebesgue, 1902)

together with sets, measures, sigma algebras and lots of machinery. The Lebesgue Integral does solve

most of the problems of the Riemann Integral at the expense of enormous complexity.


1.2. A Bit of Bias {149 words}, revised 13/10/17 at 11:30 Page 3

Then Ralph Henstock (1955) and Jaroslav Kurzweil (1957) discovered that a simple

modification to the venerable Riemann Integral produces an integral even more powerful than the

Lebesgue Integral; it includes the Newton Integral, which the Lebesgue Integral does not; and it is a

model of simplicity in definition! In their honor, we call this the “HK integral”.

The bottom line is this: For Engineers and Scientists, the HK Integral – which we view as a

simple extension of the Riemann Integral – does everything we need to do, and then some.

This note is an attempt to describe various integration mechanics in simple terms and apply them

to knotty problems. There will be more detail and proof than practical engineers are accustomed to, but

perhaps less than formal mathematicians are accustomed to.

1.2. A Bit of Bias {149 words}

I am perhaps more critical of overly complex mathematics than others. This stems in part from

my engineering and scientific background. But another ingredient is the connection between Integration

Theory and Probability Theory. It was Doob1 himself who complained that Probability Theory is “…

absurdly overloaded with extraneous mathematics.” Later in his life, Doob2 wrote “The idea that a

(mathematical) random variable is simply a function, with no romantic connotation, seemed rather

humiliating to some probabilists.” In the same paper, Doob had already noted that Kolmogorov “In the

first pages of his monograph ... states explicitly that real valued random variables are measurable (we

will soon say integrable) functions and expectations are their integrals.”

Engineers and Scientists are pragmatic. We like simplicity. We especially like to dispense with

multiple layers of complexity, which – however useful elsewhere – are unnecessary for the present task.

1.3. Some Reference Points {132 words}

We rely heavily on an excellent paper by Eric Schechter3, “An Introduction to The Gauge

Integral” that presents the HK integral in a very orderly way. He includes a short but comprehensive

bibliography. A very thorough review and taxonomy of modern integrals on the real line is given by R.

M. Dudley and R. Norvaiša4, “An Introduction to p-Variation and Young Integrals” which covers

many integral forms and has a vast bibliography. The excellent book “Elementary Stochastic

Calculus” by Thomas Mikosch5 is wonderfully brief and easy reading. Finally, the many works of

1 Doob, Joseph L., his book 2 Doob, Joseph L., “Development of Rigor in Mathematical Probability”, Development of Mathematics 1900-1950,

Birkhäuser, ISBN 3-7643-2821-5 3 Eric Schechter, http://www.math.vanderbilt.edu/~schectex/ccc/gauge/, q.v. 4 R. M. Dudley and R. Norvaiša, http://www.maphysto.dk/publications/MPS-LN/1998/1.pdf, q.v. 5 Thomas Mikosch, “Elementary Stochastic Calculus”, World Scientific, ISBN 9810235437

http://www.math.vanderbilt.edu/~schectex/ccc/gauge/

http://www.maphysto.dk/publications/MPS-LN/1998/1.pdf


2.1. What is an Integral? {219 words}, revised 13/10/17 at 11:30 Page 4

Ralph Henstock and Patrick Muldowney6 form the basis for much of the modern work about HK

integration in Rn and Function Spaces. A “Google” or “Yahoo” search will get plenty of references.

2. The Notion of Integration

2.1. What is an Integral? {219 words}

Let us get right to it. In one dimension, for integration of real valued functions over a segment of

a real line, we want (real) integral to mean all of the following three things:

1. The anti-derivative of a function

2. The area under a curve

3. The limit of a sum

The anti-derivative is called the Newton Integral in honor of the man who invented the

differential and integral calculus7. Suppose F(x) and its derivative f(x) are defined at every point on the

interval [a,b], then the Newton integral is defined to be:

x

a

dIf f (x) F(x)

dx

Then F(x) F(a) f (z)dz x [a,b]

(2.1.1)

This is surely the most popular integral we use. It does have some limitations. Chief of these is that

only a derivative of a function can be integrated.

The area under a curve is often calculated by the Riemann Integral, which is the limit of a sum

approximating “vertical stripes” as in a “trapezoid rule” approximation. The Riemann Integral is8:

Let f(x) be a real function on [a,b] and let IR be its presumed integral, then

b

Ra

I f (z)dz by definition (2.1.2)

Provided that for every >0, there exists a (real number) >0 such that

n

R k k k 1

k 1

0 1 n

k 1 k k

k k 1

I f s t t

For any partition with these three properties:

Property 1: a t t t b

Property 2: t s t

Property 3: t t

(2.1.3)

6 A good compendium of their works together with biographies and historical accounts is in: “New Integrals, Proceedings of

the Henstock Conference”, August 9-12, 1988, Springer-Verlag, ISBN 0387523227 7 Another often used name is the “Calculus Integral”. 8 This follows closely the work of Eric Schechter, ibid


2.2. Problems with the Standard Riemann Integral {91 words}, revised 13/10/17 at 11:30 Page 5

This straightforward definition (a prime example of the “epsilon-delta game”) gives an integral that is

easy to use and understand. However, there are problems.

2.2. Problems with the Standard Riemann Integral {91 words}

The Riemann Integral fails in several situations. Three examples are as follows:

1/ 2

2 2

2

x x 0Case 1: f (x) a simple unbounded function

0 x 0

1 x p / q i.e., rationalCase 2: f (x) classic "Lebesgue" example

0 x otherwise

22x cos(x ) sin(x ) x 0 d

Case 3: f (x) the notorious x cos(xxdx

0 x 0

2 )

(2.2.1)

1. Case 1: Illustrates that unbounded functions are not Riemann Integrable. This one has a simple

anti-derivative, so the Riemann Integral can’t even recover all anti-derivatives!

2. Case 2: The classic illustration of the “Power” of the Lebesgue Integral.

3. Case 3: The notorious derivative of a highly oscillatory function. It has an anti-derivative

(Newton Integral) but no Riemann or Lebesgue Integral!

There are many other examples of course.

2.3. The HK Integral Defined {187 words}

Henshaw and Kurzweil gave us a better integral with a small change to the Riemann Integral

definition in equation(2.1.3). The HK integral is:

Let f(x) be a real function on [a,b] and let IHK be its presumed integral, then

b

HKa

I f (z)dz by definition (2.3.1)

Provided that for every >0, there exists a (real function) (x)>0 such that

n

HK k k k 1

k 1

0 1 n

k 1 k k

k k 1 k

I f s t t



Property 2: t s t

Property 3: t t s

(2.3.2)


2.4. Demonstrations Using the HK Integral {462 words}, revised 13/10/17 at 11:30 Page 6

This is another “epsilon-delta game” with a single change from equation (2.1.3) above. In Property 3 of

the allowed partitions, the constant >0 is replaced by the function (x)>0.

First question: For a given (x)>0 does a “(x)–fine” partition satisfying the conditions of

equation (2.3.2) always exist? Yes, by Cousin’s Lemma, proved in Appendix A.1, q.v.

An equivalent definition of the HK integral modifies equation(2.3.2) is:

n

HK k k k 1

k 1

0 1 n

k k 1 k

k k 1 k

I f s t t



Property 2: s t , t

Property 3: t t s

(2.3.3)

The alternate definition of Property 2 constrains the evaluation points to be at the “edge” of the interval.

This change often simplifies proofs, but does not enlarge the class of integrable functions.

This innocent looking change from >0 to (x)>0 gives us an integral with more power than the

Newton Integral, the Riemann Integral and the Lebesgue Integral!!!

2.4. Demonstrations Using the HK Integral {462 words}

We will show that the HK integral fixes these three “problem cases” listed in section 2.2 above.

This demonstrated the power and simplicity of the HK Integral. We will gain some physical insight to

the application of the HK integral to problems in mathematical physics.

Case 1: The function f(x)=x-1/2

integrated over any interval including [0]. More specifically:

1/ 2

Case1

1

Case1 Case11

x x 0f (x) a simple unbounded function

0 x 0

I f (x)dx

(2.4.1)

The basic problem here for ordinary Riemann Integration is that the sum goes unbounded for intervals of

the partition near 0+. The trick is to construct a gauge (x)>0 such that the “vertical stripe width” is

suitably small near 0+. A suitable piecewise-linear (x) is:

Case1 0,Case1

1 x 0

(x) x 0 x 1

1 1 x

(2.4.2)

This (x) insures that no vertical stripe area goes infinite.


2.4. Demonstrations Using the HK Integral {462 words}, revised 13/10/17 at 11:30 Page 7

Case 2: The characteristic function of the rationals (always assumed to be in lowest terms)

integrated over (say) [-1,+1]. More specifically:

Case2

1

Case2 Case21

1 x p / q i.e., rationalf (x) classic "Lebesgue" example

0 x otherwise

I f (x)dx

(2.4.3)

The Riemann Integral fails because in each vertical stripe, it can’t figure out whether the function has

value 0 or 1. To fix this, we construct a gauge (x)>0 so that the vertical stripe is very narrow around

any rational x as follows:

(p q)

Case2 0,Case2

2 x p / q i.e., rational(x)

1 x otherwise

(2.4.4)

This fixes the problem by limiting the area of each vertical stripe to a vanishingly small number.

We can construct a similar but more challenging case, which shows the power of the HK integral

and severely tests the Lebesgue Integral, as follows:

Case 2a: The denominator function of the rationals (always assumed to be in lowest terms)

integrated over (say) [-1,+1]. More specifically:

Case2a

1

Case2a Case2a1

q x p / q i.e., rationalf (x)

0 x otherwise

I f (x)dx

(2.4.5)

The function is unbounded so both the Riemann Integral and the Lebesgue Integral fail. To fix this, we

construct a gauge (x)>0 so that the vertical stripe is very, very narrow around any rational x as follows:

(p q)

Case2a 0,Case2a

2x p / q i.e., rational

(x) q

1 x otherwise

(2.4.6)

This fixes the problem by limiting the area of each vertical stripe to a vanishingly small number. With

suitable care, we can extend this to many other cases of functions on countable sets.

Case 3: The notorious derivative of a highly oscillatory function. For more generality, we

demonstrate that the HK Integral can recover a function from its derivative.

Recall that the existence of the derivative of F(x) means that for any there is a sufficiently

small so that:

F(x h) F(x)

f (x) , For any h such that hh

(2.4.7)

This is true for every x thus there is a gauge (x)>0. With a little algebra we derive:


2.5. Some More Things about the HK Integral {334 words}, revised 13/10/17 at 11:30 Page 8

n

k k k 1

k 1

n

k k 1 k k k 1

k 1

n

k k 1 k k k 1

k 1

n

k k 1

k 1

(F(b) F(a)) f (s ) t t

(F(t ) F(t )) f (s ) t t

(F(t ) F(t )) f (s ) t t

t t

b a

(2.4.8)

(Note that that this proof is simplified by requiring sk to be either tk or tk-1.)

So the HK integral will always recover a function from its derivative. More precisely, with the

HK Integral, any function can be recovered by integration from its derivative providing the derivative

exists, and conversely (except for sets of zero measure) any function can be recovered by differentiation

from its integral provided the integral exists.

2.5. Some More Things about the HK Integral {334 words}

Here we state with little proof some additional useful characteristics of the HK integral. Some

other less interesting features are described in Appendix A.3.

1. Extension to infinite limits: Modify the definition of equations (2.3.1) and (2.3.2) as follows:

Let f(x) be a real function on [a,b] and let IHKX be its presumed integral, then

HKXI f (z)dz by definition

(2.5.1)

Provided that for every >0, there exists a (real function) (x)>0 and constants A and B such that

n

HKX k k k 1

k 1

0 1 n

k k 1 k

k k 1 k

I f s t t


Property 1: a t t t b, a A, B b

Property 2: s t , t

Property 3: t t s

(2.5.2)

This avoids taking the “limit of a limit” – always a dangerous and uncertain proposition9.

9 Other authors, like Muldowney and Henstock (himself) prefer to define () rather than use A and B as we do. The results

are equivalent.


2.6. Some Final Commentary about the HK Integral {120 words}, revised 13/10/17 at 11:30 Page 9

2. Extension to Stieltjes Form: The HK-Stieltjes Integral is defined as follows:

Let f(x) and g(x) be a real functions on [a,b] and let IHKS be its integral ∫fdg, then:

b

HKS za

I f (z)d g(z) by definition (2.5.3)


n

HK k k k 1

k 1

0 1 n

k k 1 k

k k 1 k

I f s g(t ) g(t )



Property 2: s t , t

Property 3: t t s

(2.5.4)

This is the form used by Pfeffer10 in his very nice book. The Stieltjes form integrals are used in many

places, but theorems are much harder to prove. We need it for Wiener Integrals, Stochastic Differential

Equations, and general integration with Probability Distribution Functions.

A more thorough discussion of the HK-Stieltjes Integral is contained in section 4 below.

3. The McShane Integral: An alternate and useful definition of the HK Integral is that the

interval [tk-1,tk] be in the “delta-Neighborhood” of sk, denoted by [tk-1,tk]N(,sk). Note

particularly that this does not require that sk[tk-1,tk]. Relaxing this last condition allows sk to

wander around a bit – and it yields the McShane Integral. As it turns out, this integral is

identical to the Lebesgue Integral in class of functions integrated and convergence theorems. For

those interested, this leads quickly to questions of topology, Banach (for Lebesgue and

McShane) versus Vector Spaces (for HK), etc.

4. The Hake Theorem: The Improper Riemann integral and the Improper Lebesgue Integral

increase the class of functions integrable by each. Does the Improper HK Integral increase the

class of HK integrable functions? No, by Hake’s Theorem.

2.6. Some Final Commentary about the HK Integral {120 words}

The definition of the HK integral is stunningly simple and familiar. Anyone who has done

numerical integration either by hand or by computer knows the value of variable spacing. Much of this

familiarity comes from our understanding of the real line. Extending the results to Rn presents little

problem, as Riesz-Sz.-Nagy11 elegantly demonstrate.

10 Washek F. Pfeffer, “The Riemann Approach to Integration”, Cambridge 1993, ISBN 0521440351 11 Frigyes Riesz and Bela Sz.-Nagy, “Functional Analysis”, ISBN 0486662896. This is an excellent treatment of Riemann

Stieltjes Integrals in Rn.


3.1. The Fundamental Theorem of Calculus – Anti-Derivative {158 words}, revised 13/10/17 at 11:30 Page 10

However, extending the HK integral to more abstract spaces can be problematical – especially as

compared with the Lebesgue Integral. One (quite defensive) response is this: The HK integral starts out

easy on Rn and gets harder when moving to more abstract spaces. The Lebesgue Integral starts out hard

in Rn and stays hard in more abstract spaces.

3. Important Problems Requiring Integration

3.1. The Fundamental Theorem of Calculus – Anti-Derivative {158 words}

Engineers and Scientists assume that the anti-derivative (i.e. the Newton Integral) is a proper

integral value of a derivative of almost any function – this notwithstanding the known difficulties with

the standard Riemann Integral. So, it is very comforting to find that the HK integral solves all these

problems – and more! This is a clear case of “Theory catching up with Practice”.

The paragraph above, viewed critically, illustrates much of the attitude differences between

Mathematicians and Engineer/Scientists. Much more could be said – but I will resist the temptation!

We must however continue to use common sense. Here is a classic example:

1 1

badboy 11

dtI ln(t) ln( 1) ln( 1) I

t

!! WHAT ?? (3.1.1)

This “integral” does not exist as a real integral by anybody’s definition. There are or course contour

path integrals in complex space which yield either +I or -I depending. But, on the real line, if we

ascribe any value at all, symmetry would suggest zero!

3.2. Interchanging the Order of Integrals and Limits {125 words}

There are many problems in Engineering and Scientific analysis which require the interchange of

an integral and a derivative/sum/limit as depicted in equations (1.1.2) reprised here:

b b

a a

b b

n na a

n n

b b

n na an n

d dF(z, x)dx ? F(z, x)dx

dz dz

F (z, x)dx ? F (z, x)dx

lim F (z, x)dx ? lim F (z, x)dx

(3.2.1)

A derivative is a limit. An infinite sum is a limit of a sequence of partial sums. So the problem reduces

to interchanging an integral and a limit.

The basis for most HK convergence theorems is the “Monotone Convergence Theorem” which

roughly states that:


3.3. The Wiener Integral and Machinery {409 words}, revised 13/10/17 at 11:30 Page 11

n

b b

na a

b b

na an

If: F (z, x) F(z, x)

And: F(z, x)dx and F (z, x)dx Exist for all n

Then: lim F (z, x)dx F(z, x)dx

(3.2.2)

From this, one can derive other convergence theorems (e.g., bounded, uniform, and dominated).

The bottom line again for that we do not need to worry about interchange as long as the limits

and integrals exist. (We engineers never worried much about them anyway!)

3.3. The Wiener Integral and Machinery {409 words}

The Wiener Integral is:

b

Wiener ya

I (x, ) F(x, y, )d w(y, ) (3.3.1)

This is a very general definition, in Stieltjes form12, which has many important sub-cases. However, in

every case this integral exists unconditionally whenever F(x,y,) is of Bounded Variation (BV) on y,

because w(y,) (the Wiener Process13) is continuous on y – no ifs, ands, or buts!

Several specific cases warrant discussion here, as follows:

Case1 Case1 y

x

Case2 Case2 ya

t

Case3 Case3 sa

Case 1: f (x, ) F (x y)d w(y, ) Stationary Processes

Case 2: f (x, ) F (x, y, )d w(y, ) General Volterra Form

Case 3: f (t, ) F (t,X(s, ))d w(s, ) SDE Volterra Form

(3.3.2)

Case 1: This integral is the basis for the Wiener Machinery used in the study of turbulence and

other random physical processes. We suppose that F(..) has a few derivative (and therefore is BV) and

all these are square HK integrable. With this proviso, f(x,) is stationary, “ergodic” – whenever the

limits are infinite, and its derivative is:

Case1 Case1 y

d df (x, ) F (x y)d w(y, )

dx dx

(3.3.3)

This is a terrific result for the study of turbulence. The solution process is to convert the equations of

motion (Navier-Stokes and Conservation of Mass) with dependent variables like vi(t,) to equations in

kernels like Vi(t). Such Equations, no matter how complex, are not random, and can be solved (in

principle) for once and for all. The convolution form is essential to preserve stationarity and ergodicity,

12 See section 4 below for a more detailed discussion of the Stieltjes form integrals. 13 We closely follow Wiener’s “Non-Linear Problems in Random Theory”, Technology Press MIT, Library of Congress

Catalog Number 58-59720, q.v.


3.4. Integrating Stochastic Differential Equations {361 words}, revised 13/10/17 at 11:30 Page 12

and to limit the number of free parameters. Speaking colloquially, this technique removes the

“randomness” from “turbulence”.

On a personal note, I became interested in the general theories of integration while studying

turbulence. Many authors state unequivocally that the convolution form of the Wiener Integral (Case 1

of equation(3.3.2)) does not exist because w(y,) (the Wiener Process) is not BV. Even Wiener himself

(in NPRT) uses integration by parts to justify his work – and then goes on to move derivatives inside the

integral! (Heuristic rigor indeed!) But the happy facts are that the Wiener integral exists as an ordinary

Riemann-Stieltjes Integral, or an improper one over infinite limits. Thus, we do not even need the HK-

Stieltjes form, except that the HK-integral over infinite limits is quite appealing, and moving derivatives

inside the integral is more easily justified in the infinite limit case.

Case 2: This is the general Volterra form (variable upper limit) integral. It is much studied, but

there is very little one can say in general about it.

Case 3: This is a somewhat specialized Volterra Stochastic integral arising from the integration

of the Stochastic Differential Equation (SDE) form over time:

t t

t t

d X(t, ) a(t,X(t, ))dt b(t,X(t, ))d w(t, )

presumed to be a limit of

X(t, ) a(t,X(t, )) t b(t,X(t, )) w(t, )

(3.3.4)

This is discussed in some detail in the next section and leads to extraordinary things such as the Itô and

Stratonovich integrals.

3.4. Integrating Stochastic Differential Equations {361 words}

In this section we take a quick look at SDE’s to see if the HK integral adds much to the

discussion, and to see if the techniques used are helpful in applying the Wiener Machinery. The answer

in both cases seems to be no.

The Case 3: presented in the preceding section, involves the Stochastic Differential Equation

(SDE) form14:

t td X(t, ) a(t,X(t, ))dt b(t,X(t, ))d w(t, ) (3.4.1)

This is presumed to be the limit of:

t k 1 k

v v k 1 k

v v t k 1 k

k v k 1

X(t, ) X(t , ) X(t , )

a(t ,X(t , )) ( t t t )

b(t ,X(t , )) ( w(t, ) w(t , ) w(t , ))

where: t t t

(3.4.2)

The key to this discussion is the value assigned to tv. There are three sub cases as follows:

14 This notation corresponds to Mikosch. See his Chapter 3 for many details.


3.4. Integrating Stochastic Differential Equations {361 words}, revised 13/10/17 at 11:30 Page 13

a. tv=tk, Forward Difference, Leads to Itô Integral and the Itô Lemma

b. tv=tk+1, Backward Difference, Never to my knowledge used

c. tv=tk+1/2=(tk+tk+1)/2, Central Difference, Leads to Stratonovich Integral.

There is no fundamental problem in integrating equation(3.4.1) so long as b(t,X(t,)) is BV on t,

and many important financial cases15 are in fact BV. But most geometric cases are not BV. The Itô

theory gives a very specific way of evaluating these integrals.

Focus on Case 3a: and consider the exemplar of Itô integration theory, namely the Itô Lemma:

t t

1 2 zs s

t1

222 s

2

1 2 22

f (z, w(z, ))dz f (z, w(z, ))d w(z, )f (t, w(t, )) f (s, w(s, ))

f (z, w(z, ))dz

f (a,b) f (a,b) f (a,b)where : f (a,b) f (a,b) f (a,b)

a b b b

(3.4.3)

The term involving f22(a,b) distinguishes this form from ordinary differential equations. This result is

readily obtained by expanding f(..) in a Taylor series, setting (w(t,))2 to dt, and ignoring higher order

terms. (In the Stratonovich analysis the f22(a,b) term disappears.)

Typically, these SDE’s are used in financial calculations with simple forms16 of a(..) and b(..)

such as:

1 2 1 2dX(t, ) c X(t, ) c X(t, ) dw(t, ) (3.4.4)

The full solution is very messy. A simpler case can be solved as follows:

t

t t

s0 0

dX(t, ) c X(t, )dt X(t, ) d w(t, )

X(t, ) X(0, ) c X(s, )ds X(s, ) d w(s, )

(3.4.5)

Applying the Itô lemma and guessing at the answer, we obtain:

212

X(t, ) X(0, )exp c t w(t, ) (3.4.6)

This is known as geometric (multiplicative) Brownian motion and it is the most important solution for

financial purposes. The solution can be generalized somewhat for c(t) and s(t) being function of t, as

follows:

t t

21s20 0

X(t, ) X(0, )exp c(s) (s) ds (s)d w(s, ) (3.4.7)

There are other solutions, especially for arithmetic (additive) randomness. There are other

finance models, especially log-normal models. But, aside from the brilliance of the Itô and Stratonovich

15 This is true for arithmetic (additive) randomness, especially important in the log-normal case. 16 This notation closely follows Mikosch (ibid) page 138 ff.


4.1. Riemann-Stieltjes Integration {489 words}, revised 13/10/17 at 11:30 Page 14

integrals themselves, there seems to be little added by the HK integral and little to carry over to the

Wiener Machinery. Patrick Muldowney17 has explored these issues further.

4. Riemann-Stieltjes and HK-Stieltjes Integration

4.1. Riemann-Stieltjes Integration {489 words}

The Riemann-Stieltjes Integral is very familiar to engineers and scientists. The definition is:

Let f(x) and g(x) be a real functions on [a,b] and let IRs be its integral ∫fdg, then18:

b

RS za


Provided that for every >0, there exists a (real number) >0 such that

n

RS k k k 1

k 1

0 1 n

k k 1 k

k k 1

I f s g(t ) g(t )



Property 2: s t , t

Property 3: t t

(4.1.2)

This integral is very important to fields as varied as Laplace Transforms in Electrical Engineering,

Characteristic Functions in Probability Theory, and Stochastic Theory in Wiener and Itô Integrals.

Some observations:

1. If g(x) has a bounded derivative everywhere in [a,b] then the integral becomes b b

RS za a

g(z)I f (z)d g(z) f (z) dz

z

and we revert to Riemann (or HK) Integration.

2. Useful forms for g(x) include functions with a finite (or even countable) number of steps,

providing these steps do not occur at points where f(x) is discontinuous.

3. The integral can exist when g(x) is not BV (i.e., does not have bounded variation). For

example, if g(x)=w(x,), namely, the Wiener Process, the integral is of great interest as

described in section 3 above.

There has been much confusion in the literature about the existence of this integral, so we

endeavor to describe the conditions under which this integral exists. We state the following:

1. Basic Existence19: If one of {f(x),g(x)} is continuous and the other is of Bounded Variation

(i.e., BV) then the integral b

RS xa

I f (x)d g(x) exists unconditionally.

17 Patrick Muldowney, “The Infinite Dimensional Henstock Integral and Problems of Black-Scholes Expectation”, Journal of

Applied Analysis, Vol. 8, No. 1 (2002), pp. 1-21 18 The parlance dzg(z) is used because very often we have several parameters such as dzg(x,z,t).


4.2. HK-Stieltjes Integration {144 words}, revised 13/10/17 at 11:30 Page 15

2. Integration by Parts: If b

xa

f (x)d g(x) exists for any reason, then

a. so does b

xa

g(x)d f (x) and

b. b b

x xa a

f (b)g(b) f (a)g(a) f (x)d g(x) g(x)d f (x) is the “integration by parts”

formula20.

3. Holder Formula21: Define p

p i i 1v (f ) sup abs(f (x ) f (x )) to be “p-variation” over

(say) [a,b]. (The sup is over all partitions of [a,b].) Then the integral b

xa

f (x)d g(x) exists

provided:

a. The functions f(x) and g(x) do not have discontinuities at the same point.

b. The functions f(x) and g(x) have bounded p-variation>0 and q-variation>0

respectively and 1 1p q 1

The Basic Existence condition is the one we use most often because it is critical to evaluation of

integrals with the Wiener Process as Integrator, e.g. sf (t, ) F(t s) d w(s, ) . This is because

w(t,) is continuous but not BV (of course BV means p-variation with p=1). So we cheerfully require

the F(..) be BV. This insures the existence of the Wiener Stochastic Integral. Unfortunately, much of

the literature incorrectly states that this integral does not exist because w(t,) is not BV22. The BV

restriction on F(..) is minor. We will always want F(..) to be fairly smooth and have its first few

derivatives, so F(..) necessarily has BV. Conversely, the derivative of a BV functions exists almost

everywhere23.

We only use the Integration by Parts Formula to prove completeness of Wiener Functionals.

The Holder Formula is a charming extension of the class of (RS) integrable functions that has

no application known to me.

A direct quote from Mikosch: “Weaker conditions for the existence of b

xa

f (x)d g(x) are not

very well known, but they were already found by L. C. Young in 1936; see the recent papers of Dudley

and Norvaiša (publications to appear) for an extensive discussion.”

4.2. HK-Stieltjes Integration {144 words}

The HK-Stieltjes Integral is not very familiar to engineers and scientists. The usual definition is:

19 Angus E. Taylor, “General Theory of Functions and Integration” ISBN 0-486-64988-1, Section 9-5, q.v. He proves basic

existence and Integration-by-Parts Theorems. Robert G. Bartle, “A Modern Theory of Integration” ISBN 0-8218-0845-1

proves both theorems in the context of the Henstock Integral. 20 This formula needs a little repair if either function is discontinuous at either endpoint. 21 This is directly from Mikosch, “Elementary Stochastic Calculus”, page 94. 22 Anecdotally, this fact led me on a merry search and a thorough examination of various forms of integration, including the

HK integral and the work of Dudley and Norvaiša concerning p-variation and Young Integrals. 23 See (inter alia) Wikipedia “Bounded Variation”. Related is the Riesz “Sunrise Theorem”.


5.1. The Importance of Functional Integration {337 words}, revised 13/10/17 at 11:30 Page 16

Let f(x) and g(x) be real functions on [a,b] and let IHKS be its presumed integral, then:

b

HKS za



n

HKS k k k 1

k 1

0 1 n

k k 1 k

k k 1 k

I f s g(t ) g(t )



Property 2: s t , t

Property 3: t t (s )

(4.2.2)

Some observations:

1. The class of f(x) which can be integrated against a given g(x) is greatly increased over the

traditional Riemann-Stieltjes Integral and includes all three cases mentioned in section 2.2

for smooth enough g(x). Gordon and/or Pfeffer have many theorems on this subject.

2. If g(x) is BV, it can be split into two monotone non-decreasing functions such that

g(x)=g+(x)-g

-(x). This simplifies many proofs.

For us, the main value of using the HK-Stieltjes form is that the integration limits [a,b] can be

extended to infinity with no additional limit process.

5. Integration in Function Spaces

5.1. The Importance of Functional Integration {337 words}

The discussion above concerns integration in R (the real numbers). We can readily extend these

results to real vector spaces Rn. However, integration in Function Spaces or Functional Integration is an

esoteric topic whose main result is to rigorously define and justify the Wiener Path Integral and the

Feynman Path Integral.

The Wiener Path Integral is of great theoretical importance in probability theory. It is a

foundation block for the Wiener Process, Stochastic Integration, Wiener Functionals, and the whole of

the “Wiener Machinery” as Yaglom calls it. As usually presented, the Wiener Integral can be rigorously

justified using standard Riemann Integration with extensions to infinite limits24. The HK Integral makes

this process much easier and it yields a somewhat broader result.

The Feynman Path Integral is of equally great theoretical importance in quantum mechanics

because it brings robustness to Feynman’s path integrals. As usually presented, the Feynman integral

24 Wiener shows this (with typical “heuristic” rigor, per Yaglom) in NPRT.


5.2. The Mathematics of Functional Integration {299 words}, revised 13/10/17 at 11:30 Page 17

cannot be rigorously justified using standard Riemann or even Lebesgue integration methods25.

Muldowney and (separately) Henstock have developed a method of functional integration based on the

real HK integral that rigorously justifies the Feynman integral.

As we shall see, this integration mechanism puts the Wiener Path Integral and the Feynman Path

Integral on the same footing because each is a specific case of a single combined integral with different

values of a complex parameter. This is a very important theoretical result.

This is the theory as usually presented. But I claim that both the Wiener and the Feynman Path

integrals can be evaluated exactly, given the usual definition of path integrals as the limit of an n-fold

integration. The extension of the HK-integral to functional integrals (by decomposable division spaces)

may perhaps lead to a wider class of functions that have path integrals, but this has little effect on the

physical problems under study. On the other hand, one rarely evaluates a path integral for a practical

result – an exception is numerical evaluation of Wiener Functionals for visualization purposes.

5.2. The Mathematics of Functional Integration {299 words}

Following Stevens26, A functional integral is the limit of an n-fold integral:

n 1

kk 1n 0 1 n

f (x) n

k k

n 0 1 n

dfG f (x) f (x) lim (n 1) g (f , f , , f )

f f (x )

P a x x x b a "partition"

D

(5.2.1)

For convergence, we require that:

1. The integration is carried out over the Range of f(x), usually (-,+).

2. There is a partition Pn which is “refined” in the limiting process.

3. The n-parameter function gn(f1,…

,fn) converges to the functional G[f(x)].

4. The parameter is a normalizing factor insuring that the integral converges.

As an example, consider the functional integral

t 2

x([ ,t ])

y(t) 1K x,t; , : exp y (s) ds

2

D

(5.2.2)

The integration is conducted over all paths y(t) which begin y= and end at y(t)=x(t). The

normalizing parameter is chosen to make K(..) a proper probability density, i.e. its integral over all x

is 1. The traditional method for evaluating such an integral is as follows:

1. Construct a partition n 0 1 nP (t) t t t t

2. Perform each of the n-fold integrations analytically or otherwise.

25 The tone of this statement telegraphs that I respectfully, but pointedly disagree, read on. 26 Charles F. Stevens, “The Six Core Theories of Modern Physics”, Bradford, ISBN 0262193590


5.3. The Wiener Path Integral {189 words}, revised 13/10/17 at 11:30 Page 18

3. Allow n to increase without bound while the mesh size of the partition goes to zero.

But now we need to think a bit about the meaning of Functional Integration. Suppose that for

every n, we can analytically evaluate the n-fold integral of equation(5.2.1). Suppose we can show

convergence of the resulting real or complex result as n increases without bound. We can in fact do this

precisely for both the Wiener Path Integral (using the error function) and the Feynman Path Integral

(using Fresnel Integrals).

Does this resolve all the issues of Functional Integration for these two cases? The answer

depends on one’s point of view. If we accept Steven’s definition of functional integration (most of us

do), then we must also accept the convergence of cylindrical functions like. gn(f1,…

,fn)) to the functional

G[f(x)]. Steven’s also claims that this convergence is assured for continuous functionals (such as G[..])

of continuous functions (such as f(x)).

5.3. The Wiener Path Integral {189 words}

The most frequently encountered kernel is the Wiener Kernel, which represents the Wiener

Process. Think of this as a “Random Walk” in time increments of t yielding incremental space

movement of x. The Wiener Process is (of course) a Markov Process, so the increments are

independent. We therefore formulate the functional integral as:

2

1/ 2

n 1 n

k k k k-1 k-1k 1 k 1n

2

n k k 1

k 1k k 1n 1

k n 1/ 2k 1nk k 1k 1

x1 1w x,t; , exp

2 (t )2 (t )

p x,t; , lim dx w x ,t ;x ,t

x x1exp

2 (t t )lim dx

2 (t t )

(5.3.1)

An important detail is that the upper limit of the dxk product is only n-1, whereas the tk product and

the exponential sum have upper limits of n.

As it turns out, this multiple integral can be evaluated analytically by considering first the

integral for dxn-1 which can be evaluated exactly, and continuing n-2, etc. back to k=1. The result is the

Wiener probability density function w(x,t;) so that

n 1 n

k k k k-1 k-1k 1 k 1n

2

1/ 2

p x,t; , lim dx w x ,t ;x ,t

x1 1w x,t; , exp

2 (t )2 (t )

(5.3.2)


5.4. The Feynman Path Integral {192 words}, revised 13/10/17 at 11:30 Page 19

This result for the Wiener Path Integral is not too revealing. The answer is precisely as expected, and

the algebra required to evaluate the n-fold integrals is not burdensome. The importance of the result is

more theoretical, because it shows the internal consistency of the Wiener Kernel and justifies the rest of

the machinery for constructing Wiener Measure.

5.4. The Feynman Path Integral {192 words}

The Feynman Path Kernel represents a quantum process and a solution to the Schrödinger

equation. Think of this as a “Propagation” of a particle in time increments of t resulting in incremental

space “movement” of x. The quantum propagator is a Markov Process, so the increments are

independent. The Feynman Path Integral is the limit of the n-fold integral:

2

1/ 2

n 1 n

k k k k-1 k-1k 1 k 1n

2

n k k 1

k 1k k 1n 1

k n 1/ 2k 1nk k 1k 1

xIwf x,t; , 2 I(t ) exp

2 (t )

pf x,t; , lim dx wc x ,t ;x ,t

x xIexp

2 (t t )lim dx

2 I(t t )

(5.4.1)

An important detail is that the upper limit of the dxk product is only n-1, whereas the tk product and

the exponential sum have upper limits of n.

As before, this multiple integral can be evaluated analytically by considering first the integral for

dxn-1 which can be evaluated exactly, and continuing n-2, etc. back to k=1. The result is the Feynman

propagator function wf(x,t;) so that

n 1 n

k k k k-1 k-1k 1 k 1n

2

1/ 2

pf x,t; , lim dx wf x ,t ;x ,t

x1 Iwf x,t; , exp

2 (t )2 I(t )

(5.4.2)

Again, the answer is precisely as expected, and the algebra required to evaluate the n-fold integrals is

harder than the Wiener case, but it is not too burdensome. The importance of the result is more

theoretical, because it shows the internal consistency of the Feynman Kernel and justifies the rest of the

machinery for constructing the Schrödinger equation.


5.5. Outline of Muldowney’s Work on Functional Integration {462 words}, revised 13/10/17 at 11:30 Page 20

5.5. Outline of Muldowney’s Work on Functional Integration {462 words}

Muldowney27 reworked the Wiener Path Integral using HK integration techniques. He enlarged

the notion of “partition” to a collection of -fine partitions on the xk that depend on each other as well as

the particular partition on t. He then applied this technique to a more general kernel of the form28

21/2

12

12

xwc x,t; , (t ) exp c

c (t )

c Yields Wiener Paths

c I Yields Feynman Paths

(5.5.1)

In the preceding section, we evaluated the Wiener Path Integral and the Feynman Path Integral as

convergent limits of n-fold integrations of an exponential kernel. However, the results are not entirely

satisfactory. Each integral is defined as a limit, and if the Improper Riemann (or Lebesgue) integral is

required, then another limit process is required. In all there are at least n cascaded limits, perhaps as

many as 2n+1 cascaded limits.

Muldowney approached the functional integral problem as a single limit. He constructed a

“decomposable division space” which contains collections of divisions that in turn generate convergent

sums approximating the integral. Each division consists of an integer n, an n-element partition of time,

and a finite collection of n -fine partition29 of the xk. For each division there is a defined sum that

approximates the integral. Then comes the epsilon-delta game: The value of the integral is V, if, given

any >0, there exists a collection of divisions such that abs(V-Sum)<, where Sum is computed using

any member of the collection. This effectively reduces the limit process to a single limit. Because of

this, several useful convergence theorems can be proved.

Comparing Muldowney’s method to the Integrals of section 5 is much like comparing HK-

integration with standard Riemann Integration. We can now rigorously move derivatives and limits

under the integral sign – critical to deriving the Diffusion and Schrödinger equations. Moreover, the

class of functional forms which can be integrated is somewhat increased.

A final similarity that Scientists and Engineers appreciate is this – we never get into trouble

using the simple methods, because the Muldowney-Henstock machinery stands squarely behind us.

Muldowney feels very strongly about the virtues of these division spaces. He writes:

“In the opinion of this writer, the three most productive and distinctive ideas in integration theory are

division, variation and decomposability. For, while partitions have always been used in integration,

Henstock has shown the fundamental importance of divisions. Also, the variation is widely used in

27 Patrick Muldowney, “A general theory of integration in function spaces”, 1987, Pitman Research Notes in Mathematics

Series" 28 The complete Feynman kernel has a term analogous to potential energy. This term presents no problem to establishing the

existence of the path integral. 29 As it turns out, each partition must be an HK-gauge to make the division space decomposable.

http://www.academia.edu/220630/A_general_theory_of_integration_in_function_spaces


6.1. The Role of Measure Vs. Integration {338 words}, revised 13/10/17 at 11:30 Page 21

integration theory but it is especially useful in non-absolute integrations such as the Henstock generalized

Riemann and function space integrations.

“And, regarding decomposability, Henstock (12 “Additivity …”) has emphasized that it is this property of

integration systems, rather than countable additivity of the measure, that enables us to prove limit theorems

such as the dominated convergence theorem. Thus, limit theorems of this kind are available in generalized

Riemann but not in Riemann integration.”

6. Integration in Probability Spaces

6.1. The Role of Measure Vs. Integration {338 words}

Prologue: This is a short discussion of Random Variables as functions of [0,1] and expected

values as integrals yielding expected values30. This defines probability rather than measures of sets.

Classical Probability Theory defines a Probability Space as the triple {,F,P}. These are:

1. is the Sample Space. For example, an element might be a pointer to a sample path

of Brownian motion, as w(t,).

2. F is a sigma algebra of P measurable subsets (events) of For example, an element of F

might be the set of all Brownian motion paths greater than z at t0, i.e. {w(t0,)>z}.

3. P is a probability measure of members of F.

Then a real Random Variable is a function mapping to the real line, e.g. f:R.

Often omitted from the definition are the following considerations:

1. the Sample Space has the cardinality of the real line (i.e. 1), and thus might as well be

[0,1]

2. F is not all subsets of , which would have the cardinality of the power set of the real line

(i.e. 2). Instead, F must have the cardinality of the real line. It must also be a sigma

algebra to insure countable additivity necessary for Lebesgue integration.

3. The measure P must allow Lebesgue integration over , and the integral value is the

expected value. Thus: F f ( )P(d )

or simply 1

0F f ( )d if

Working backward, the Lebesgue integral forces us to use a “sigma algebra” (vs. a simple algebra) and

it limits Random Variables to be those which are absolutely integrable.

A more modern basis for Probability Theory expands the class of sets in F, does not require

countable additivity, and uses HK integration to define expected values. For example, consider the

famously oscillatory function

30 A much more detailed analysis is contained in my document “Pretty Good Probability Theory”, q.v.


6.2. A Simple Probability Theory Model {107 words}, revised 13/10/17 at 11:30 Page 22

2 2 2 2

1

0

2 df ( ) 2 cos( ) sin( ) cos( )

d

f ( ) f ( )d cos(1)

E (6.1.1)

This is a proper RV in the HK theory and impossible with the standard Lebesgue theory. It illustrates

the Muldowney contention that decomposability is a more basic notion to integration and probability

theory than is countable additivity.

6.2. A Simple Probability Theory Model {107 words}

The model suggested above has been explored by both Wiener and Muldowney, and also Doob,

Kolmogorov, Lumley, Yaglom, and me. Some have credited Hugo Steinhaus with the model, since he

proposed it much before the seminal Kolmogorov paper.

There seems to be no reason to use anything other than =[0,1]. Expected values are defined by

HK integrals over [0,1], and probability is defined in terms of characteristic functions of subsets of F.

We do not much care whether F is a “sigma algebra” or not. As a practical matter, no physicist ever

pays attention to the properties of F.

Appendix A: Some Extra Materials on Integration

A.1. Cousin’s Lemma {213 words}

We restate the definition of the HK Integral (from equations (2.3.1) and (2.3.2)):

Let f(x) be a real function on [a,b] and let IHK be its presumed integral, then

b

HKa

I f (z)dz by definition (A.1.1)


n

HK k k k 1

k 1

0 1 n

k k 1 k

k k 1 k

I f s t t



Property 2: s t , t

Property 3: t t s

(A.1.2)

For arbitrary (x)>0, does such a “(x)-fine” partition meeting the requirements stated in

equation (A.1.2) actually exist? Yes, as proved by Cousin’s Lemma.

One proof is by contradiction as follows:


A.2. Some Thoughts about Delta-Fine Partitions {145 words}, revised 13/10/17 at 11:30 Page 23

Assume there is no such (x)-fine partition. Construct a sequence {IN} of sub intervals of [a,b]

of length 2-n

. For each n, select the leftmost interval that is not (x)-fine, and call it Inkn. This

sequence of intervals is nested and shrinks to zero length; hence (by the nested interval theorem)

there is a common point in them all, (say) z. But (z) is strictly greater than zero by hypothesis,

so there is some n such that Inkn is (z)-fine, contradicting the assertion.

An alternate proof (shorter, but harder) is as follows:

Let Set x (x) 2,x (x) 2 : x a,b , then is an infinite open covering of a

bounded closed interval. By the Heine-Borel Theorem, there is a finite covering subset YΨ.

Remove the overlaps, and close the resulting collection to get a (x)-fine partition of [a,b].

A.2. Some Thoughts about Delta-Fine Partitions {145 words}

Consider the arbitrary function (x)>0. Suppose that (x) has a lower bound inf((x)>0. Then

there is an ordinary Riemann Partition for this (x), and we gain nothing. In general the greatest power

of the HK Integral comes when inf((x)=0.

Now suppose that inf((x)=0. Can (x) be a continuous function? It certainly need not be

continuous, but must it fail continuity? I think so, but I can’t prove it right now (131017). I recall an

article saying that (x) can be made to be “left semi-continuous” – whatever that means!

It is worth noting that a general function like (x) defined on [a,b] has the cardinality of the

power set of R, (i.e. 2). Such functions exist even when bounded or restricted to a range of (say) [1,2].

Such functions could be valid gauge functions, but little is gained.

A.3. Some More Thoughts about the Integration {178 words}

There are a number of topics which are related to integration methods, but which have had little

coverage in this sketchy note. They include (in no particular order):

1. Regulated Functions: Used by Dudley and Norvaiša, essentially functions with limits in both

directions at every point.

2. Young Partitions: Collections of non-overlapping open intervals and singletons which exhaust

[a,b]. Removes a number of discontinuity problems in Stieltjes form integrals.

3. Proof of Stieltjes Integration by Parts theorem: Essence of the proof comes from this

rearrangement of the sum n n

k k k 1 n n 1 1 0 k k 1 k

k 1 k 1

f g g f g f g g f f

. If the right side

sum converges, then so does the left. Granted some cleanup at the ends.

4. Mesh vs. Refinement integrals: Per D&N, define “Refinement Partitions” which give slightly

better results than regular Riemann (Mesh) partitions.

5. Change of variable: Many oscillatory integrals can be tamed with transformation such as x=1/t.

6. Convergence Factors: Over infinite domains especially, multiplying the kernel by a

convergence factor like exp(-ax2) and taking the limit as a0 can tame a number of integrals.


A.4. Details of Some Feynman-Fresnel Type Integrals {383 words}, revised 13/10/17 at 11:30 Page 24

A.4. Details of Some Feynman-Fresnel Type Integrals {383 words}

The following is a lot of tedious algebra to evaluate some special Fresnel style integrals. The

general form is:

x

2p 2 2p 2

0 0

I(p, x) : z exp i z dz, I(p) : z exp i z dz2 2

(A.4.1)

These I(p) integrals are very oscillatory and are certainly not absolutely integrable. However, for p<½,

they are (as we shall see) “Improper Riemann Integrable” and HK integrable. If we allow an “Improper

Lebesgue Integral” with the obvious definition, then they are also Lebesgue integrable.

Consider first the case for p=0 and write:

0 2i20

2 2 2i2 2 20

I(0) : z exp z dz

exp z cos z i sin z dz

(A.4.2)

Actually, these are the standard Fresnel Integrals31, namely:

x x2 2

2 20 0C(x) : cos z dz, S(x) : sin z dz

1 1C( ) S( )2 2

(A.4.3)

However, we are trying to prove convergence, so we proceed as follows. Consider first the

sin(..) case – the treatment for cos(..) is quite similar. Then:

2

220 0

4(k 1)

k 24k

k 0

IS(0) : sin z dz sin t 2 t dt

D : sin t 2 t dt

(A.4.4)

Stated this way, the sin(..) function is integrated through successive full cycles. The trick is to

find a tight bound on the Dk and prove convergence of the sum. Thus:

31 You can’t beat the NBS “Handbook of Mathematical Functions” for things like this.



4(k 1)

k 24k

4k 2 4(k 1)

2 24k 4k 2

4k 2 4k 4

2 2

4k 4k 2

D : sin t 2 t dt

sin t 2 4k dt sin t 2 4(k 1) dt

1 2 1 2cos t cos t

4 k 4 k 1

1 4 1 4 1 1 1 k 1 k

4 k 4 k 1 k k 1 k(k 1)

k 1 k k(k 1) 1

k(k 1) kk 1 k

3/ 2

k(k 1) 1

(k 1) 2 k 2 k k 1

k

2

(A.4.5)

So what is this series?

3/ 2 3/ 2 1/ 2

km 1

k m k m m 1

1 1 1 1D k s ds s

2 2 m 1

(A.4.6)

This proves the convergence of the integral given by equation(A.4.4).

This is a very interesting integral result. The kernel oscillates rapidly and does not tend to zero

as t. The detail that makes the sums converge is that the positive half waves nearly cancel the

negative half waves.

To explore the matter a little further, suppose that 0p<½, then equation(A.4.5) becomes:

4(k 1)p 1/ 2

k 24k

4k 2 4(k 1)p 1/ 2 p 1/ 2

2 24k 4k 2

1/ 2 p

1/ 2 p

1/ 2 p 2

D (p) : t sin t 2dt

(4k) sin t 2dt (4k 4) sin t 2dt

2 11 1

(4k) k

2 1 11 1 1/ 2 p 1/ 2 p 1/ 2 p 1

(4k) 1!k 2!k

1 2p

2 (4)

(3/ 2 p)

pk

(A.4.7)

Then the sum corresponding to equation(A.4.6) becomes:



(3/ 2 p) (3/ 2 p)

k p p m 1k m k m

(1/ 2 p)

p p 1/ 2 p

m 1

1 2p 1 2pD (p) k s ds

2 (4) 2 (4)

1 2p 1 1s

2 (4) (1/ 2 p) (4) (m 1)

(A.4.8)

This converges for all 0p<½ .

If p=0, the result is as before. If p>0, we get another extraordinary integral result. The kernel

oscillates rapidly and grows without bound as t! The sums converge because the positive half

waves cancel the negative half waves more completely than the tp term grows – mirabile dictu.

Another technique uses a “convergence factor”. Modify the integral in equation(A.4.4) to be:

2p 2 2 p

220 0

4(k 1)p

k 24k

k 0

IS(p, ) : z sin z exp z dz t sin t exp t 2 t dt

D : t sin t exp t 2 t dt

(A.4.9)

The idea here is that ultimately the exponential will overwhelm the power term and the sinusoid to force

convergence. Then the limit as 0 is computed. If this limit exists and is finite then this is the value

of the integral.

Finally consider the integral:

n

20IX(n, ) : z sin exp( z) dz, 0, n 0

(A.4.10)

Change variables to yield:

n

20

n

2

1

IX(n, ) : z sin exp( z) dz, 0, n 0

dtt exp( z) dt exp( z)dz dz

t

ln(t) dtIX(n, ) : sin t

t

(A.4.11)

This gives us a convergent Improper Riemann integral of an unbounded highly oscillatory function.



Local Table of Contents:

1. The Task to Be Accomplished ................................................... 2

1.1. A Little History and Background {493 words} .....................................................2

1.2. A Bit of Bias {149 words} .....................................................................................3 1.3. Some Reference Points {132 words} .....................................................................3

2. The Notion of Integration ........................................................... 4

2.1. What is an Integral? {219 words} ..........................................................................4 2.2. Problems with the Standard Riemann Integral {91 words} ...................................5 2.3. The HK Integral Defined {187 words} ..................................................................5 2.4. Demonstrations Using the HK Integral {462 words} ............................................6

2.5. Some More Things about the HK Integral {334 words} .......................................8

2.6. Some Final Commentary about the HK Integral {120 words} ..............................9

3. Important Problems Requiring Integration ............................. 10

3.1. The Fundamental Theorem of Calculus – Anti-Derivative {158 words} ............10 3.2. Interchanging the Order of Integrals and Limits {125 words} ............................10 3.3. The Wiener Integral and Machinery {409 words} ..............................................11

3.4. Integrating Stochastic Differential Equations {361 words} ................................12

4. Riemann-Stieltjes and HK-Stieltjes Integration ...................... 14

4.1. Riemann-Stieltjes Integration {489 words} .........................................................14 4.2. HK-Stieltjes Integration {144 words} .................................................................15

5. Integration in Function Spaces ............................................... 16

5.1. The Importance of Functional Integration {337 words} .....................................16 5.2. The Mathematics of Functional Integration {299 words} ...................................17

5.3. The Wiener Path Integral {189 words} ...............................................................18 5.4. The Feynman Path Integral {192 words} ............................................................19

5.5. Outline of Muldowney’s Work on Functional Integration {462 words} .............20

6. Integration in Probability Spaces ............................................ 21

6.1. The Role of Measure Vs. Integration {338 words} .............................................21 6.2. A Simple Probability Theory Model {107 words} ..............................................22

Appendix A: Some Extra Materials on Integration ..................... 22

A.1. Cousin’s Lemma {213 words} ............................................................................22

A.2. Some Thoughts about Delta-Fine Partitions {145 words} .................................23 A.3. Some More Thoughts about the Integration {178 words} ..................................23 A.4. Details of Some Feynman-Fresnel Type Integrals {383 words} ........................24

Appendix B: Comments 131016 .................................................. 27

Appendix B: Comments 131016

1. A possible writing rule: “Contrary to the convention in most modern mathematical exposition

and the wishes of the GTM editors, I often use the first person singular rather than the “royal we"

when I expect the reader to be playing a passive role. I restrict the use of “we" to places, like

proofs, where I expect the active participation of my readers.”

(7300 words)

Pretty Good Integration Theory - WordPress.com...The excellent book “Elementary Stochastic Calculus” 5by Thomas Mikosch is wonderfully brief and easy reading. Finally, the many

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