Generalisation of Vitali's covering lemma

Question

In "The geometry of fractal sets", Falconer gives the following generalisation of the Vitali covering lemma as an exercise:

Let $\mu$ be any measure on $\mathbb{R}^{n}$ and $E$ a set with $\mu$-finite measure. If $\mathcal{V}$ is a Vitali class of measurable sets for $E$, then one can find disjoint sets $U_{1}, U_{2},... \in\mathcal{V}$ such as $\mu(E\backslash\bigcup U_{i})=0$.

($\mathcal{V}$ is said to be a Vitali class for $E$ if for all $x \in E$ and $\varepsilon > 0$ there is a $U$ in $\mathcal{V}$ such as $x\in U$ and $diam(U) < \varepsilon$.)

I was quite surprised by this result, because there is no topological hypothesis whatsoever on $\mu$. I've been working on this exercise for two or three days and haven't really come close to proving it or disproving it. If you have some ideas I would be very grateful.

Thank you very much

Answer 1

I haven't thought about Vitali type results in many years, so I cheated and asked someone I know who has done research in related topics.

The result as stated is easily seen to be false by letting $\mathcal{V}$ be the family of finite sets and $\mu$ be Lebesgue measure.

Less trivial counterexamples can probably be found by using (for the sets that make up $\mathcal{V})$ suitable families of rotated rectangles in the plane.

(ADDED NEXT DAY) This morning I looked at my copy of Falconer's book and saw that I had made a comment by Exercise 1.4, saying there was a problem with the exercise and I cited p. 444 of a review by Gulisashvili. I looked up the review in my "book review collection" and brought it with me so that I could post the relevent stuff when I had a chance.

Archil B. Gulisashvili, [Review of Falconer's The Geometry of Fractal Sets], Leningrad Mathematical Journal 2 #2 (1991), 439-447. [This is an English translation by Harold Hogan McFaden of the original Russian version that appeared in Algebra i Analiz 2 #2 (1990), 249-259.]

What follows is a direct quote from the English translation, near the bottom of p. 444.

It should be mentioned that in Exercise 1.4 of Chapter 1 Vitali's theorem for general measures is not stated correctly. An extra condition on the measure or on the covering family was apparently omitted. It is not hard to prove that for $0 < \alpha < 1$ the restriction of ${\mathscr H}^{\alpha}$ to some set $C_{\alpha}$ of Cantor type on the line $R^1$ does not satisfy the conclusion of Vitali's theorem for a specially chosen covering family, although all the conditions of Exercise 1.4 are satisfied. The covering theorem in Exercise 1.4 is later used in Exercise 1.8, but here everything is true, because one of the known variants of Vitali's theorem can be applied under the assumptions of Exercise 1.8.