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href="#fb3_img_img_13fb278d-787b-57dd-84ba-c495502d8176.png" alt="integral Underscript bold upper R Superscript upper T Endscripts f"/> has two components: domain , and integrand . When is an infinite set, such as an interval of time, two different perspectives are present. These are the perspectives indicated in figures 3.1 and 3.2 on page 87 of [MTRV].

      Figure 3.1 represents the graph of

where is an argument of integrand . Figure 3.2 represents the domain whose elements are points, not graphs. Both of these perspectives should be kept in mind while using this book.

To see the significance of these alternate perspectives¹, suppose

and , so . In this case, for integrands which appear in this book, there may be little difference between the values of integrands in domain , and in if is continuous.

      But domains

and are very different; just as differs geometrically from , . The latter difference is the vehicle for the 19th century satire Flatland by Edwin A. Abbott [1], in which two‐dimensional beings struggle with the idea of a three‐dimensional universe.

      The book can be read as a collection of standalone accounts of topics which are suggested, or introduced, or touched upon, in [MTRV]. Equally, it can be read as a supplement to [MTRV], developing the ideas of stochastic sums and path integrals which were introduced in [MTRV].

      The formal theorem‐proof layout is avoided in favour of concrete illustrations. In almost cases, the relevant theorems and proofs are cited from [MTRV]. But exceptions are made for a few fundamental results such as analogues or versions of theorems 4, 62, 64, and 65 of [MTRV]. The theorem‐proof format is applied in these few exceptional cases.

      Also, some of the illustrative examples are “thought experiments”, exercises of imagination intended to clarify, not finance or physics as such, but certain mathematical issues which arise in these subjects.

      Notes and additional material for the book can be found at:

      

https://sites.google.com/site/StieltjesComplete/

      —also used to record typos, errors, and corrections; and cited as [ website ] in this book.

      Sincere thanks to colleagues and readers who help by sharing their thoughts via:

[email protected]

Note

      1 ¹ In [MTRV], usually denotes , with denoting . Also, is used in [MTRV] to denote , or .

Introduction

      The gauge integral is a version of the Riemann integral, with much improved convergence properties. Convergence properties are conditions which ensure integrability of a function; in particular, integrability of the limit of a convergent sequence of integrable functions, with integral of the limit equal to the limit of the integrals—the limit theorems.

      Another notable property of the gauge integral

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