1. Recall that we defined the Borel σ-algebra in \( \mathbb{R}^n \), to be the σ-algebra generated by the collection of open boxes, i.e.,\[\mathcal{B}^n := \sigma(\{(a_1, b_1) \times (a_2, b_2) \times \cdots \times (a_n, b_n) \mid a_i < b_i, a_i, b_i \in \mathbb{R} \text{ for all } i\}).\](a) Let \( n = 1 \). Formally show that all the sets of the form \( (-\infty, x], \, (-\infty, x), \, [x, \infty), \, (x, \infty), \, (x, \, x), \, \{x\}, \, [x, y] \) belong to \( \mathcal{B}^1 \) for any \( x, y \in \mathbb{R} \) with \( x \leq y \) (we sketched the proofs in the class).(b) For \( n = 2 \), show that the following sets are Borel-sets in \( \mathbb{R}^2 \): \( (-\infty, x] \times (-\infty, y], \, (-\infty, x) \times (-\infty, y), \, [x, \infty) \times [y, \infty), \, [x_1, y_1] \times [x_2, y_2] \) belong to \( \mathcal{B}^2 \) for any \( x, y, x_1 \leq y_1, x_2 \leq y_2 \in \mathbb{R} \).(c) Show that the open ball of radius 1 around zero, i.e.,\[D_1(0) := \{ (x, y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1 \},\]is also in \( \mathcal{B}^2 \).Hint: use the fact that set of points in \( \mathbb{R

Since you have posted a question with multiple subparts, we will solve the first two subparts…

Recall that we defined the Borel o-algebra in R", to be the o-algebra generated by the collection of open boxes, i.e., B" := 0({(a1,b1) × (a2, b2) × · . · x (an, bn) | a, < bi, ai , by E R for all i}). (a) Let n = 1. Formally show that all the sets of the form (-0, 2], (-0, 2), [r, o0), (x, o0), {r}, [r, y) belong to B' for any r, y ER with r < y (we sketched the proofs in the class). (b) For n = 2, show that the following sets are Borel-sets in R²: (-00, 2] x (-oco, y], (-0, a) x (-0, y), [r, o0) x [r, o0), {(r, y)}, [T1, Y1] x [r2, y2] belong to B² for any r, y, x1 < y1, 12 € y2 € R.

Recall that we defined the Borel o-algebra in R", to be the o-algebra generated by the collection of open boxes, i.e., B" := 0({(a1,b1) × (a2, b2) × · . · x (an, bn) | a, < bi, ai , by E R for all i}). (a) Let n = 1. Formally show that all the sets of the form (-0, 2], (-0, 2), [r, o0), (x, o0), {r}, [r, y) belong to B' for any r, y ER with r < y (we sketched the proofs in the class). (b) For n = 2, show that the following sets are Borel-sets in R²: (-00, 2] x (-oco, y], (-0, a) x (-0, y), [r, o0) x [r, o0), {(r, y)}, [T1, Y1] x [r2, y2] belong to B² for any r, y, x1 < y1, 12 € y2 € R.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

1. Recall that we defined the Borel σ-algebra in \( \mathbb{R}^n \), to be the σ-algebra generated by the collection of open boxes, i.e.,

\[
\mathcal{B}^n := \sigma(\{(a_1, b_1) \times (a_2, b_2) \times \cdots \times (a_n, b_n) \mid a_i < b_i, a_i, b_i \in \mathbb{R} \text{ for all } i\}).
\]

(a) Let \( n = 1 \). Formally show that all the sets of the form \( (-\infty, x], \, (-\infty, x), \, [x, \infty), \, (x, \infty), \, (x, \, x), \, \{x\}, \, [x, y] \) belong to \( \mathcal{B}^1 \) for any \( x, y \in \mathbb{R} \) with \( x \leq y \) (we sketched the proofs in the class).

(b) For \( n = 2 \), show that the following sets are Borel-sets in \( \mathbb{R}^2 \): \( (-\infty, x] \times (-\infty, y], \, (-\infty, x) \times (-\infty, y), \, [x, \infty) \times [y, \infty), \, [x_1, y_1] \times [x_2, y_2] \) belong to \( \mathcal{B}^2 \) for any \( x, y, x_1 \leq y_1, x_2 \leq y_2 \in \mathbb{R} \).

(c) Show that the open ball of radius 1 around zero, i.e.,

\[
D_1(0) := \{ (x, y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1 \},
\]

is also in \( \mathcal{B}^2 \).

Hint: use the fact that set of points in \( \mathbb{R

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