Question 5. We endow [0, 1] with the Lebesgue 1-dimensional measure X¹ and the Borel o-algebra B([0, 1]). We endow [0, 1] x [0, 1] with the product o-algebra B(R) B(R) and the product measure A¹ A¹. Note: the product measure space considered above, that is ([0, 1] × [0, 1], B([0, 1]) B([0, 1]), X¹ X¹) is the same as ([0, 1] × [0, 1], B([0, 1] x [0, 1]), A²), where X2 is the Lebesgue 2-dimensional measure. You will not need this fact. = Let f [0, 1] → R be B([0, 1])-measurable. Consider g: [0, 1] x [0, 1] → R defined by g(x, y) f(x). (a) Prove that g is measurable with respect to B([0, 1]) B([0, 1]). Assume moreover that g is integrable on [0, 1] x [0, 1] with respect to X¹ X¹. (b) Prove that f is integrable on [0, 1] with respect to X¹. (c) Prove that [0,1]x[0,1] 9d(X¹ X¹) = [0,1] ƒ dX¹.
Question 5. We endow [0, 1] with the Lebesgue 1-dimensional measure X¹ and the Borel o-algebra B([0, 1]). We endow [0, 1] x [0, 1] with the product o-algebra B(R) B(R) and the product measure A¹ A¹. Note: the product measure space considered above, that is ([0, 1] × [0, 1], B([0, 1]) B([0, 1]), X¹ X¹) is the same as ([0, 1] × [0, 1], B([0, 1] x [0, 1]), A²), where X2 is the Lebesgue 2-dimensional measure. You will not need this fact. = Let f [0, 1] → R be B([0, 1])-measurable. Consider g: [0, 1] x [0, 1] → R defined by g(x, y) f(x). (a) Prove that g is measurable with respect to B([0, 1]) B([0, 1]). Assume moreover that g is integrable on [0, 1] x [0, 1] with respect to X¹ X¹. (b) Prove that f is integrable on [0, 1] with respect to X¹. (c) Prove that [0,1]x[0,1] 9d(X¹ X¹) = [0,1] ƒ dX¹.
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 5. We endow [0, 1] with the Lebesgue 1-dimensional measure X¹ and the Borel o-algebra
B([0, 1]). We endow [0, 1] x [0, 1] with the product o-algebra B(R) B(R) and the product measure
X¹ X¹.
Note: the product measure space considered above, that is ([0, 1] × [0, 1], B([0, 1]) & B([0, 1]), X¹0
X¹) is the same as ([0, 1] × [0, 1], B([0, 1] × [0, 1]), A²), where X2 is the Lebesgue 2-dimensional
measure. You will not need this fact.
Let f [0, 1] → R be B([0, 1])-measurable. Consider g: [0, 1] x [0, 1] → R defined by g(x, y) =
f(x).
(a) Prove that g is measurable with respect to B([0, 1]) B([0, 1]).
Assume moreover that g is integrable on [0, 1] x [0, 1] with respect to X¹ X².
(b) Prove that f is integrable on [0, 1] with respect to X¹.
(c) Prove that [0,1]x[0,1] 9 d(X¹ X¹) = [0,1] ƒ dX¹.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7c4cf19a-236e-4fca-aa5c-a3680ecd3848%2F159d35af-a3ea-4dd7-8ea6-f8809e5e66e9%2Fctuauid_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 5. We endow [0, 1] with the Lebesgue 1-dimensional measure X¹ and the Borel o-algebra
B([0, 1]). We endow [0, 1] x [0, 1] with the product o-algebra B(R) B(R) and the product measure
X¹ X¹.
Note: the product measure space considered above, that is ([0, 1] × [0, 1], B([0, 1]) & B([0, 1]), X¹0
X¹) is the same as ([0, 1] × [0, 1], B([0, 1] × [0, 1]), A²), where X2 is the Lebesgue 2-dimensional
measure. You will not need this fact.
Let f [0, 1] → R be B([0, 1])-measurable. Consider g: [0, 1] x [0, 1] → R defined by g(x, y) =
f(x).
(a) Prove that g is measurable with respect to B([0, 1]) B([0, 1]).
Assume moreover that g is integrable on [0, 1] x [0, 1] with respect to X¹ X².
(b) Prove that f is integrable on [0, 1] with respect to X¹.
(c) Prove that [0,1]x[0,1] 9 d(X¹ X¹) = [0,1] ƒ dX¹.
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