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
Related questions
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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F833481a2-df8c-4805-95a2-f24b64ba619f%2F2132bc35-ced8-4dad-a982-8e41d4fa2124%2Flxtt97d_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
