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Let 2 = R and F be the o-algebra generated by A = Q (i.e., F = o({Q})). (a) Write explicitly all the sets belonging to F. (b) Characterize all the mappings X : 2 R that are random variables with respect to (2, F)

Let 2 = R and F be the o-algebra generated by A = Q (i.e., F = o({Q})). (a) Write explicitly all the sets belonging to F. (b) Characterize all the mappings X : 2 R that are random variables with respect to (2, F)

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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**Problem 214 of Prof. Kim's Notes** Let \( \Omega = \mathbb{R} \) and \( \mathcal{F} \) be the \(\sigma\)-algebra generated by \( A = \mathbb{Q} \) (i.e., \( \mathcal{F} = \sigma(\{\mathbb{Q}\}) \)). (a) Write explicitly all the sets belonging to \( \mathcal{F} \). (b) Characterize all the mappings \( X : \Omega \rightarrow \mathbb{R} \) that are random variables with respect to \( (\Omega, \mathcal{F}) \).
Transcribed Image Text:**Problem 214 of Prof. Kim's Notes** Let \( \Omega = \mathbb{R} \) and \( \mathcal{F} \) be the \(\sigma\)-algebra generated by \( A = \mathbb{Q} \) (i.e., \( \mathcal{F} = \sigma(\{\mathbb{Q}\}) \)). (a) Write explicitly all the sets belonging to \( \mathcal{F} \). (b) Characterize all the mappings \( X : \Omega \rightarrow \mathbb{R} \) that are random variables with respect to \( (\Omega, \mathcal{F}) \).
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Step 1

It is given that Ω=ℝ, and F is the σ-algebra generated by A=Q.

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