Examples of textbooks that use $\overline{\overline{A}}$ to denote set cardinality?

Question

I was recently introduced to the notation $\overline{\overline{A}}$ to refer to the cardinality of a set $A$. (The dashes above the set of interest might be more closely spaced together.)

In the hundreds of books I have, I don't have a single textbook that uses this notation to refer to the cardinality of a set. What are some examples of books that do? (I'd be especially interested in one that covers measure theory, but this isn't necessary.)

As mentioned in the comments, Wikipedia also cites this notation as an example.

This is horrible notation... where did you come across it? $|A|$ is standard. — Qiaochu Yuan, Sep 21 '20 at 19:58
@QiaochuYuan My professor uses it, and you can also see Wikipedia. — Clarinetist, Sep 21 '20 at 19:59
All I can hear in my head is swear words looking at that notation. Bar notation is commonly used for closure of a set in a topological space. This is just terrible notation and that is being kind. What on earth. — Cameron L. Williams, Sep 21 '20 at 19:59
I think $\overline{\overline{\overline{\overline{\underline{\underline{A}}}}}}$ makes more sense — David P, Sep 21 '20 at 20:01
@DavidP Agree. Although it can be confused with $\underline{\underline{\overline{\overline{\overline{A}}}}}$ which has a thoroughly different meaning. A worthy price for elegance and restraint in notation. — Olivier Moschetta, Sep 21 '20 at 20:07
@QiaochuYuan: It’s an old notation. Patrick Suppes uses it in his Axiomatic Set Theory (1960). For what it’s worth, he says that it stems from Cantor’s use of the double bar to indicate two levels of abstraction. I think that I’ve seen it in a few other older sources, too. — Brian M. Scott, Sep 21 '20 at 20:10
The conclusion is: up to middle '70s it was the "standard" notation. — Mauro ALLEGRANZA, Sep 22 '20 at 09:41

Dave L. Renfro · Accepted Answer · 2020-10-13T09:43:55.470

I looked through some of the books I have where I thought it might be used and found it in the following books. The years given are for the edition in front of me, and the page numbers given are where I think it first appears in the book, which may not be where you’ll see it most used in the book.

Axiomatic Set Theory by Patrick Suppes (1960, p. 109)

Cardinal and Ordinal Numbers by Wacław Sierpiński (1965, p. 136)

Set Theory by K. Kuratowski and A. Mostowski (1968, p. 174)

The Theory of Sets and Transfinite Arithmetic by Alexander Abian (1965, p. 361)

Introduction to Set Theory and Topology by Kazimierz Kuratowski (1962, p. 61)

Abstract Set Theory by Abraham A. Fraenkel (1961, p. 60)

The Theory of Sets and Transfinite Numbers by B. Rotman and G. T. Kneebone (1966, p. 99)

Abstract Set Theory by Thoralf A. Skolem (1962, p.3)

Introduction to Axiomatic Set Theory by Jean-Louis Krivine (1971, p. 23)

Introduction to the Foundations of Mathematics by Raymond L. Wilder (1965, p. 104)

Introduction to Metamathematics by Stephen Cole Kleene (1952, p. 9)

Introduction to Mathematical Logic by Elliott Mendelson (1979, p. 2)

The Mathematics of Metamathematics by Helena Rasiowa and Roman Sikorski (1963, p. 12)

Sets, Lattices, and Boolean Algebras by James C. Abbott (1969, p. 65)

Boolean Rings by Alexander Abian (1976, p. 9)

Real and Abstract Analysis by Edwin Hewitt and Karl Stromberg (1969, p. 19)

Theory of Functions of a Real Variable (Volume I) by I. P. Natanson (1961, p. 27)

The Theory of Functions of Real Variables by Lawrence M. Graves (1956, p. 308)

The Theory of Functions of a Real Variable and the Theory of Fourier’s Series (Volume I) by E. W. Hobson (1927, p. 203)

You can add Thoralf Skolem, Abstract Set Theory (1962), page 3 as well as Jean-Louis Krivine, Introduction to Axiomatic Set Theory (Engl.transl.1971, Springer), page 23. — Mauro ALLEGRANZA, Sep 22 '20 at 09:31
Also Helena Rasiowa & Roman Sikorski, The Mathematics of Metamathematics (Warsawa, 1963), page 12 — Mauro ALLEGRANZA, Sep 22 '20 at 09:40
@Mauro ALLEGRANZA: The Rasiowa/Sikorski book has been on my "buy sometime" list for many years, and has recently risen close enough to the top that probably the next time I purchase some books from amazon, I'll try to get a reasonably decent copy of it. I know about the other two books (have checked them out from university libraries several times in the past 40+ years), but I don't have them and they're not on my "list". Of course, if I actually saw one of them at a used bookstore (unlikely), I'd buy it. I'll add these books to the list later, because there might be others suggested. — Dave L. Renfro, Sep 22 '20 at 09:45

Examples of textbooks that use $\overline{\overline{A}}$ to denote set cardinality?

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