psie
- 315
- 40
- TL;DR Summary
- I know the dyadic cubes in ##\mathbb R^d## generate the Borel ##\sigma##-algebra of ##\mathbb R^d##. Let now ##U\subset \mathbb R^d## be an open subset and the claim is that all dyadic (half-open) cubes whose closure are contained in ##U## generates the Borel ##\sigma##-algebra on ##U##. How?
Dyadic cubes of order ##n\in\mathbb N## are sets of the form $$C=\prod_{j=1}^d (k_j2^{-n},(k_j+1)2^{-n}],\quad k_j\in\mathbb Z.$$ I know that the dyadic cubes in ##\mathbb R^d## induce a generating collection for the Borel ##\sigma##-algebra of ##U## (by intersecting the dyadic cubes in ##\mathbb R^d## with ##U##), however, this induced generating collection may be larger than merely the dyadic cubes whose closure is in ##U##.
If I can somehow show that every open subset of ##U## is a countable union of dyadic cubes whose closure is in ##U##, then I'm done. Note, an open subset of ##U## is of the form ##A\cap U##, where ##A## is open in ##\mathbb R^d## (so the open subsets of ##U## are open in ##\mathbb R^d## as well). I feel stuck on how to proceed.
If I can somehow show that every open subset of ##U## is a countable union of dyadic cubes whose closure is in ##U##, then I'm done. Note, an open subset of ##U## is of the form ##A\cap U##, where ##A## is open in ##\mathbb R^d## (so the open subsets of ##U## are open in ##\mathbb R^d## as well). I feel stuck on how to proceed.