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I would appreciate some help understanding a step on the proof of Luzin's Theorem on Sheldon Axler's Measure, Integration & Real Analysis (open access here, Theorem 2.91 pg 66).

Basically, there is a finite collection of disjoint Borel sets $D_1, \dots, D_n$, colsed sets $F_k \subset D_k$ and open sets $G_k\supset D_k$

Then, he defines a set

$$F=\left(\bigcup_{k=1}^n F_k\right)\cup \left(\bigcap_{k=1}^n \mathbb{R}\setminus G_k\right)$$

And then claims that $\mathbb{R}\setminus F = \bigcup_{k=1}^n (G_k \setminus F_k)$. I cannot see how this follows, unless $G_i \cap F_j= \emptyset$ for $i\neq j$, and I don't see why this would be the case.

Many thanks!

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  • $\begingroup$ You are right $\mathbb{R}\setminus F = \bigcup_{k=1}^n (G_k \setminus F_k)$ is not necessarily true. Thankfully this is not a problem as only the inclusion is needed i.e. $\mathbb{R}\setminus F \subset \bigcup_{k=1}^n (G_k \setminus F_k)$ $\endgroup$ – clark Apr 6 '20 at 18:43
  • $\begingroup$ Thanks! The fact that it was not necessary for the proof went way over my head :) $\endgroup$ – SuperModular Apr 6 '20 at 19:23
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You are correct; the $=$ in the equation in your question below the displayed equation should be $\subset$. Only the set inclusion is used in the proof, so everything else should be fine. I will correct this typo in the next printing of the book. Thank you for pointing out this typo.

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    $\begingroup$ Thank you for taking the time to answer, Professor Axler! And thank you for the book, and for publishing it on open access. It is an amazing resource! $\endgroup$ – SuperModular Apr 7 '20 at 10:13

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