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During my project, I have to deal with Baire class 1 function and Baire class 2 function. I never have them in my study before. I did some research but did not bring too much. Actually, I am looking for basic properties for these classes and more.

Question: Do you know reference lecture note/book/paper for these class?

Edit: The OP was missing a very important word in the question. I meant Baire class one function and Baire class 2 function.

Thank you in advance.

00GB
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2 Answers2

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(expansion of list of references originally given in comments)

See the following Stack Exchange questions/answers:

Examples of Baire class 2 functions

See also this 4 May 2010 sci.math post and the links in it.

Construction of a function which is not the pointwise limit of a sequence of continuous functions

Baire class 1 and discontinuities

Is there a different name for strongly Darboux functions?

Is every Lebesgue measurable function on $\mathbb{R}$ the pointwise limit of continuous functions?

Examples of dense sets in the complex plane

Maximum Baire class of a Riemann integrable function

What are examples of real-valued functions of Baire class at least 3? (see also this paper)

For the first explicit constructions of a Baire $3$ function that isn’t Baire $2,$ and a Baire $4$ function that isn’t Baire $3,$ and even (for every finite positive integer $n)$ a Baire $n+1$ function that isn’t Baire $n,$ see this 14 May 2009 sci.math post, which I later incorporated (with more bibliographic detail) into my answer to Example of function of Baire 3 and Baire 4.

How could we use the continuum hypothesis if we are able prove its truth?

Mikhail Y. Suslin and Lebesgue's error — 2 sci.math posts, both made 29 July 2006: 1st post and 2nd post

For books that deal with Borel sets and Baire functions from a real analysis perspective (and thus are not heavily steeped with axiomatic set theoretic concerns), see my answer to Is there any good text introducing a part of Borel-hierarchy which is in need in measure theory.

Also of possible use are the following 4 Masters theses:

[1] Lester Randolph Ford, Pointwise Discontinuous Functions, Master of Arts Thesis (under Earle Raymond Hedrick), University of Missouri (Columbia, Missouri), 1912, iii + 45 + 14 pages.

[2] Ralph Lee Bingham, Borel Sets and Baire Functions, Master of Arts Thesis (under William Mackie Myers), Montana State University (Bozeman, Montana), 1962, iii + 84 pages.

[3] Glen Alan Schlee, On the Development of Descriptive Set Theory, Master of Arts Thesis (under Richard Daniel Mauldin), University of North Texas (Denton, Texas), August 1988, iii + 66 pages.

[4] Frank Anthony Ballone, On Volterra Spaces, Masters Thesis (under Zbigniew Piotrowski), Youngstown State University (Ohio, USA), June 2010, vi + 76 pages.

Dave L. Renfro
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Nowadays most people call "sets of the first class" meagre, and "sets of the second class" nonmeagre, so those terms might help you look for more information on the topic.

As for more specific references, it depends on what about them you want to know.

For applications to functional analysis (almost always through the baire category theorem), you should look in Chapter $5$ of Folland's Real Analysis or Chapter $2$ of Rudin's Functional Analysis. For applications to analysis that require slightly fewer prerequisites, Munkres's Topology Chapter $8$ is quite good.

Since you mention descriptive set theory in the tags, you might be more interested in definability properties. I'm a big fan of Tserunyan's lecture notes for this, and honestly the section on baire category is worth reading no matter what applications you're interested in. For more than you ever wanted to know, you should look at Chapter $I.8$ in Kechris's Classical Descriptive Set Theory.

I hope this helps ^_^

HallaSurvivor
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  • HallaSurvivor. Thank you, Sorry I missed a very important word in my question. I edit now. – 00GB Jun 13 '21 at 09:22
  • I have no idea why your answer got a downvote. Upvoting . . . – Dave L. Renfro Jun 13 '21 at 12:40
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    Nice references, but I think "Baire class 1 and 2" means something totally different from "first and second category": https://en.wikipedia.org/wiki/Baire_function – Alex Kruckman Jun 13 '21 at 14:33
  • @AlexKruckman, As you can see in the edit. The OP was missing the word Baire class one function and Baire class two function. – 00GB Jun 13 '21 at 14:40
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    @Alex Kruckman: For some reason I didn't realize HallaSurvivor was conflating Baire category classes with Baire function classes! Baire category is so tied up with Baire functions that, in my too-quick scan of the answer, I mentally "filled in the blanks" without realizing it. FYI, for Baire category one usually thinks of the first two Baire function classes, but a little known result of Lebesgue from 1905 gives an analogy of the Baire class 1 result for all the higher classes — see pp. 165−166 in Medvedev's Scenes from the History of Real Functions. – Dave L. Renfro Jun 13 '21 at 15:31
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    @AlexKruckman -- Oh wow! That's what I get for answering questions late at night. Deleting this now. – HallaSurvivor Jun 13 '21 at 18:42
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    Apparently you can't delete an accepted answer... The more you know. – HallaSurvivor Jun 13 '21 at 18:44
  • @HallaSurvivor, your answer was useful this why I accepted. Thank you so much – 00GB Jun 13 '21 at 19:53