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I am recently going to make a series of videos about real analysis and measure theory. I wonder if anyone can give me some suggestions on how to arrange the material of the course. Should I introduce abstract concepts(such as measure, $\sigma$-field, metric space, etc.)first, or can I start with some interesting examples?

My intended audience includes those who have already studied basic analysis course and know something about theoretical physics.

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    $\begingroup$ Could you list a few of those interesting examples? That might help the advice you get here. Are these examples meant to surprise students about new concepts? Or show them something they didn't know about familiar items (subsets of real numbers)? Something else? $\endgroup$ – Nick C Jan 16 '18 at 23:29
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    $\begingroup$ What background are you expecting your intended viewer to have? $\endgroup$ – David Steinberg Jan 16 '18 at 23:46
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    $\begingroup$ Personally, I find that if I haven't taught a class before, the best strategy (for me) is to find a book that I like, and parallel their approach. As I become more familiar with the strengths and weaknesses of that approach, I'll make change where I see fit. In that spirit, I might recommend Royden, but suggest that you stay away from Folland (though, to be fair, I like Folland's approach, just not the pathological terseness). McDonald and Weiss is also an option, as they are more example-forward (though you might want to trim the redundancy of chapter 4). $\endgroup$ – Xander Henderson Jan 17 '18 at 3:23
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    $\begingroup$ It may be foolish to make a series of videos like this if you have never taught the course to real live students. $\endgroup$ – Gerald Edgar Jan 17 '18 at 13:29
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    $\begingroup$ @GeraldEdgar But bear in mind that NOT everyone has a chance to teach live students. $\endgroup$ – Ma Joad Jan 17 '18 at 22:11
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This comment from MSE by Kimchi Lover is fantastic.

Examples every time. Can you think of a target end example result and work the course up to understanding it? If I were teaching it, I'd make the goal something like "Brownian motion paths are continuous and nowhere differentiable" or "most numbers are normal numbers" and construct measure theory and real analysis to be able to state & prove this rigorously.

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I do not know of any systematic treatment of the curricular sequencing issues for real analysis. James Propp wrote an article that discusses different axiomatic approaches to real analysis and the pedagogical implications of each. Lara Alcock also has a book chapter about mathematics pedagogy and real analysis that discusses pedagogical strategies as they relate to the content of the course. Another good source for ideas on presenting the material could be William Bauldry's real analysis textbook for prospective secondary mathematics teachers.

Alcock, L. (2010). Interactions Between Teaching and Research: Developing Pedagogical Content Knowledge for Real Analysis. In Learning Through Teaching Mathematics (pp. 227–245). Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3990-3_12

Bauldry, W. C. (2011). Introduction to Real Analysis: An Educational Approach. John Wiley & Sons. Preview available at: https://books.google.com/books?hl=en&lr=&id=eClOhyJmxDkC&oi=fnd&pg=PR7&dq=%22real+analysis%22+pedagogy&ots=h8GulTVPTb&sig=Cbky5PZp4y2Wg3O1f7cOOwNzsk8#v=onepage&q=%22real%20analysis%22%20pedagogy&f=false

Propp, J. (2013). Real Analysis in Reverse. American Mathematical Monthly, 120(5), 392–408. https://doi.org/10.4169/amer.math.monthly.120.05.392

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