Are there any calculus textbooks that mention Lebesgue or measure theory?
I know some undergraduate analysis books (like Baby Rudin) has its last chapter about Lebesgue theory but that book already is called "Principles of Mathematical Analysis" so the book wouldn't be used for a calculus course.
I'm looking for a book that (1) is about calculus but (2) directly references analysis topics and what kind of courses come after calculus classes.
Most often the books give a vague idea of "this topic will be seen in more advanced/future/later courses" but there's nothing explicitly mentioned.
Solution 1:
Gerald Folland's Advanced Calculus I used for the Indiana University Analysis Tier 1 qualifying Exam, which expected you to know calculus topics from an analytic perspective. It does exactly what you want in terms of being "about calculus" but directly referencing analysis topics.
Ch. 1 is where it references the analysis topics, and in Ch. 4 there is a brief section on Lebesgue Measure and Lebesgue Integrals, and it also talks about "rectifiable sets" in multiple dimensions.
Solution 2:
If, as you say, you are looking for a book that is ostensibly about calculus but makes direct reference to topics more often considered in an analysis course I would recommend you take a look at:
The Calculus Integral by Brian S. Thomson (2010).
From the preface of the book, one finds:
Our purpose is to present integration theory at an honours calculus level and in an easier manner by defining the definite integral in a very traditional way, but a way that avoids the equally traditional Riemann sums definition. Riemann sums enter the picture, to be sure, but the integral is defined in the way that Newton himself would surely endorse. Thus the fundamental theorem of the calculus starts off as the definition and the relation with Riemann sums becomes a theorem (not the definition of the definite integral as has, most unfortunately, been the case for many years). As usual in mathematical presentations we all end up in the same place. It is just that we have taken a different route to get there. It is only a pedagogical issue of which route offers the clearest perspective. The common route of starting with the definition of the Riemann integral, providing the then necessary detour into improper integrals, and ultimately heading towards the Lebesgue integral is arguably not the best path although it has at least the merit of historical fidelity.
Towards the end of the text the Lebesgue integral is introduced, and even the Henstock-Kurweil integral makes a small appearance.
The book can be downloaded for free from the author's website here. Alternatively, a traditional paper bound copy can be had for a very reasonable price.
Solution 3:
You might give a look to A Course in Advanced Calculus, by Robert S. Borden.
I'm looking for a book that (1) is about calculus but (2) directly references analysis topics and what kind of courses come after calculus classes.
The whole chapter 8 is about measure and integration, including the Lebesgue integral. From here the description of the book:
This remarkable undergraduate-level text offers a study in calculus that simultaneously unifies the concepts of integration in Euclidean space while at the same time giving students an overview of other areas intimately related to mathematical analysis. The author achieves this ambitious undertaking by shifting easily from one related subject to another. Thus, discussions of topology, linear algebra, and inequalities yield to examinations of innerproduct spaces, Fourier series, and the secret of Pythagoras. Beginning with a look at sets and structures, the text advances to such topics as limit and continuity in $E^n$, measure and integration, differentiable mappings, sequences and series, applications of improper integrals, and more. Carefully chosen problems appear at the end of each chapter, and this new edition features an additional appendix of tips and solutions for selected problems.
Here I attach the contents:
