# Proof of Luzin's Theorem on Axler's Measure, Integration & Real Analysis

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!

• 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. $\mathbb{R}\setminus F \subset \bigcup_{k=1}^n (G_k \setminus F_k)$ – clark Apr 6 '20 at 18:43
• Thanks! The fact that it was not necessary for the proof went way over my head :) – SuperModular Apr 6 '20 at 19:23

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.