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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 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
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.
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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