Question

Part b.)

2. Let Bn be the ơ-algebra of all Borel sets in Rn and .Mn be the-algebra of all the measurable sets in Rn (a) Define Bn x Bk the a-algebra generated by "Borel rectangles" Bi x B2 with Bi E Bn and B2 E Bk. Prove that Bn x BB+k (b) Does a similar result hold for measurable sets, i.e. is MnXM-Mn+A? Here Mn x M is a σ.algebra generated by "Lebesgue rectangles" L1 ×し2 with Li E M" and L26 Mk.

Answer 1

Given let B_n be the sigma algebra at all Barce sets in R^n. let M_n be the sigma algebra at all measurable sets R^n. Define B_n or B_k the sigma algebra generated by Borel rectangles as B_1 times B_2. With B_1 elementof B_n and B_2 elementof B_K. Whose B_n be the algebra at all barel aets in R^n and B_k be the sigma algebra at all Barel sets in R^n claim: - B_n times B_k = B_n + k A sigma algebra is condition unions, countable intersection and compute as B_n be sigma = algebra of a Borel sets in R^n. i. e: in the nth dimension. And B_K is sigma-algebra at all Borel sets in R^k i.e: in the kth dimension. \r\n B_k is sigma algebra of an Barel sets in n^k let B_1 elementof B_n, B_2 elementof B_k. Then formed Barel rectangle is B_1 times B_2. This B_1 times B_2 Barel rectangle generates the