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Distribution Modulo One and Diophantine Approximation by Yann Bugeaud

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Appendix C

Measure theory

In this appendix we recall some basic facts from measure theory. We begin with the easy half of the Borel–Cantelli lemma and the definition of the Hausdorff dimension. Then, we define the standard measure on the middle third Cantor set and establish some of its useful properties. We conclude with a few words on ergodic theory.

C.1 The easy half of the Borel–Cantelli lemma

We start with an easy and well-known lemma, often referred to as the (easy half of the) Borel–Cantelli lemma. Since Cantelli pointed out that the total independence of the events was not needed in the proof of Lemma C.1, the next lemma should perhaps be called the Cantelli lemma.

LEMMA C.1. Let S be a set equipped with a measure μ. Let (Ej)j≥1 be a ...

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