Denote by \(B_{r}(x)\) an open ball in \(\mathbb{R}^{n}\) with centre \(x\) and radius \(r\). Show that the Borel sets \(\mathscr{B}\left(\mathbb{R}^{n}ight)\) are generated by all open balls \(\mathbb{B}:=\left\{B_{r}(x): x \in \mathbb{R}^{n}, r>0ight\}\). Is this still true for the family \(\mathbb{B}^{\prime}:=\left\{B_{r}(x): x \in \mathbb{Q}^{n}, r \in \mathbb{Q}^{+}ight\}\)?

[mimic the proof of Theorem 3.8]

Data from theorem 3.8

Transcribed Image Text:

Theorem 3.8 We have B(R") = o(J) = o(J) = o(J") = o(gon). Proof We begin with open rectangles having rational endpoints. Since the open rectangle (a, b)= X(a,, b;) is an open set [], we get the following inclusions: 0(0) o(J) o(at). B (x) U le J Conversely, if UEO, we have U= U I IEJICU Here is clear from the definition and for the other direction 'C' we fix some x = U. Since U is open, there is some ball Be (x) CU-see Fig. 3.2- and we can inscribe a square into Be (x) and then shrink this square to get a rectangle I = I'(x) = at containing x. Since every rectangle is uniquely determined by its main diagonal, there are at most #(Q" x Q") = #N many I in the union (3.3). Thus Fig. 3.2. The ball B. (x) CU. UE OCO(Jat), proving the other inclusion o(0) Co(at), and so o(0) = o(J) = 0 (Jat). Every half-open rectangle (with rational endpoints) can be written as (3.3) [a, b) ... [an, bn) = N(a - b ) ... (an-7, bn), EN while every open rectangle (with rational endpoints) can be represented as (c, d) ...x (en, dn) = U[c + , d) ... [cn + 1, dn). x EN These formulae imply that co(o) and Joco(J) [resp. Jrat C o(at) and Jat Co(rat)], hence, by Remark 3.5(iii), o (J) = o(J) [resp. o(at) = o(Irat)] and the proof follows as we already know that o(at)= 0(ga)=0(0).