Question

. Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a general probability space, and suppose a random variable $X$ on this space is measurable with respect to the trivial $\sigma$-algebra $\mathcal{F}_0=\{\emptyset, \Omega\}$. Show that $X$ is not random (i.e., there is a constant $c$ such that $X(\omega)=c$ for all $\omega \in \Omega)$. Such a random variable is called degenerate.

    . Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a general probability space, and suppose a random variable $X$ on this space is measurable with respect to the trivial $\sigma$-algebra $\mathcal{F}_0=\{\emptyset, \Omega\}$. Show that $X$ is not random (i.e., there is a constant $c$ such that $X(\omega)=c$ for all $\omega \in \Omega)$. Such a random variable is called degenerate.

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Stochastic Calculus for Finance II : Continuous-Time Models

Stochastic Calculus for Finance II : Continuous-Time Models

Steven E. Shreve 1st Edition

Chapter 2, Problem 1

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Step 1

A random variable $X: \Omega \to \mathbb{R}$ is measurable with respect to a $\sigma$-algebra $\mathcal{F}$ if for every Borel set $B \subseteq \mathbb{R}$, the preimage $X^{-1}(B) = \{\omega \in \Omega : X(\omega) \in B\}$ is in $\mathcal{F}$. Show more…

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. Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a general probability space, and suppose a random variable $X$ on this space is measurable with respect to the trivial $\sigma$-algebra $\mathcal{F}_0=\{\emptyset, \Omega\}$. Show that $X$ is not random (i.e., there is a constant $c$ such that $X(\omega)=c$ for all $\omega \in \Omega)$. Such a random variable is called degenerate.

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