**1.0 Introduction**

Every teacher, I guess, of an introductory or intermediate course has a struggle with how to teach material that requires more advanced concepts, outside the scope of the course, to fully understand. I think it would be nice for the textbooks not to foreclose the possibility of pointing out this requirement. I here provide a couple of examples from some mathematics textbooks that I happen to have.

**2.0 Probability**

Hogg and Craig (1974) is a book on probability and statistics. I have found many of their examples and theorems of use in a wide variety of areas. They usually do not explain how many of these ideas can be expanded to an entire applied course.

**2.1 Borel Sets and Non-Measurable Sets**

An axiomatic definition of probability is a fundamental concept for this book. Hogg and Craig recognize that they do not give a completely rigorous and general definition:

Let

denote the set of every possible outcome of a random experiment; that is,Cis the sample space. It is our purpose to define a set functionCP(C) such that ifCis a subset of, thenCP(C) is the probability that the outcome of the random experiment is an element ofC...

Definition 7:IfP(C) is defined for a type of subset of the space, and if,C

P(C) ≥ 0,- Let
Cbe the union ofC_{1},C_{2},C_{3}, ... ThenP(C) =P(C_{1}) +P(C_{2}) +P(C_{3}) + ..., where the setsC_{i},i= 1, 2, 3, ..., are such that no two have a point in common...P() = 1,Cthen

P(C) is called theprobability set functionof the outcome of the random experiment. For each subsetCof, the numberCP(C) is called the probability that the outcome of the random experiment is an element of the setC, or the probability of the eventC, or the probability measure of the setC.

Remark.In the definition, the phrase 'a type of subset of the space' would be explained more fully in a more advanced course. Nevertheless, a few observations can be made about the collection of subsets that are of the type... -- Hogg and Craig (1974): pp. 12-13 (Notation changed from original).C

**2.2 Moment Generating and Characteristic Functions**

Hogg and Craig work with moment generating functions throughout their book. In the chapter in which they introduce them, they state:

Remark:In a more advanced course, we would not work with the moment-generating function because so many distributions do not have moment-generating functions. Instead, we would letidenote the imaginary unit,tan arbitrary real, and we would defineφ(t) =E(e^{itX}). This expectation exists foreverydistribution and it is called thecharacteristic functionof the distribution...Every distribution has a unique characteristic function; and to each characteristic function there corresponds a unique distribution of probability... Readers who are familiar with complex-valued functions may write

φ(t) =M(it) and, throughout this book, may prove certain theorems in complete generality.Those who have studied Laplace and Fourier transforms will note a similarity between these transforms and [the moment generating function]

M(t) andφ(t); it is the uniqueness of these transforms that allows us to assert the uniqueness of each of the moment-generating and characteristic functions. -- Hogg and Craig (1978): pp. 54-55.

**3.0 Fourier Series**

Lin and Segel (1974) provides a case study approach to applied mathematics. They introduce certain techniques and concepts in the course of specific problems. Fourier analysis is introduced in the context of the heat equation. They then look at more generals aspects of Fourier series and transforms. They state:

Suppose that we now pose the following problem, which can be regarded as the converse to Parseval's theorem. Given a set of real numbers

a_{0},a_{m},b_{m},m= 1, 2, ..., such that the series(1/2)a_{0}^{2}+ {[a_{1}^{2}+b_{1}^{2}] + [a_{2}^{2}+b_{2}^{2}] + ...}is convergent, is there a function

f(x) such that the series(1/2)a_{0}+ {[a_{1}cos(x) +b_{1}sin(x)]+ [a_{2}cos(2x) +b_{2}sin(2x)] + ...}is its Fourier series?

An affirmative answer to this question depends on the introduction of the concepts of Lebesque measure and Lebesque integration. With these notions introduced, we have the

Riesz-Fisher theorem, which states that(i) the [above] series ... is indeed the Fourier series of a function f, which is square integrable, and that (ii) the partial sums of the series converge in the mean to f.The problem we posed is a very natural one from a mathematical point of view. It appears that it might have a simple solution, but it is here that new mathematical concepts and theories emerge. On the other hand, for physical applications, such a mathematical question does

notarise naturally. -- C. C. Lin & L. A. Segel (1974): p. 147 (Notation changed from original).

**4.0 Discussion**

Here is a challenge: point out such candid remarks in textbooks in your field. I suspect many can find such comments in many textbooks. I will not be surprised if some can find some in mainstream intermediate textbooks in economics. Teaching undergraduates in economics, however, presents some challenges and tensions. I think of the acknowledged gap between undergraduate and graduate education. Furthermore, I think some tensions and inconsistencies in microeconomics cannot be and are never resolved in more advanced treatments. Off the top of my head, here are two examples.

- The theory of the firm requires the absence of transactions costs for perfect competition to prevail. But under the conditions of perfect competition, firms would not exist. Rather workers would be independent contractors, forming temporary collectives when convenient.
- Under the theory of perfect competition, as taught to undergraduates, firms are not atomistic. Thus, when taking prices as given, the managers are consistently wrong about the response of the market to changes they may each make to the quantity supplied. On the other hand, when firms are atomistic and of measure zero, they do not produce at the strictly positive finite amount required by the theory of a U-shaped average cost curve.

My archives provide many other examples of such *tensions*, to phrase it nicely.

**References**

- Robert V. Hogg & Allen T. Craig (1978).
*Introduction to Mathematical Statistics*, 4th edition, MacMillan. - C. C. Lin & L. A. Segel (1974).
*Mathematics Applied to Deterministic Problems in the Natural Sciencs*, Macmillan

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