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Let $(Ω,B,P)$ be $([0,1],B[0,1],λ)$ where is the Lebesgue measure on $[0,1]$. Define the process $\{X_t,0≤t≤1\}$ by

$$X_t(\omega)=\begin{cases}0,\text{ if }t\neq\omega\\ 1,\text{ if }t=\omega\end{cases}$$

show that each $X_t$ is a random variable. What is the $σ$-field generated by $\{X_t,0≤t≤1\}$?

Note: $B$ is a Borel set and $B[0,1]=σ(C[0,1])$

I confused on this question.( like this: $([0,1],B[0,1],λ)$...) And I could not solve it.

Thanks a million.

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What is $C[0,1]$? – Ashok Nov 18 '11 at 10:23
Do you know how to find the $\sigma$-field generated by a single $X_t$? – Ilya Nov 18 '11 at 10:41
sorry,I don't know how to find this. – eric chen Nov 18 '11 at 10:55
C[0,1] means subinterval of [0,1] . – eric chen Nov 18 '11 at 10:57
up vote 1 down vote accepted

Ok, so you should do the following. Some intuition first: $\Omega = [0,1]$ means that you randomly pick up a point $\omega\in[0,1]$ such that the probability $$ \mathsf {Pr}\{\omega\in (a,b)\} = \lambda ((a,b)) = b-a $$ for any interval $(a,b)\subseteq [0,1]$. As a result, you obtain the stochastic process $X_t$ which is zero everywhere but at a randomly chosen point $\omega$ where it is equal to one.

That was intuition, now let us comment: you don't need the probability measure here since all you ask about is the measurability which is completely described by the shape of $X_t$ and $(\Omega,\mathscr B([0,1]))$.

Now, to find the $\sigma$-algebra $\mathscr F$ generated by $(X_t)_{0\le t\le 1}$ we consider each $X_t$ separately to start with. So, $$ X_t:\Omega\to \mathbb R $$ is just a function. To derive $\sigma$-algebra $\mathscr F_t$ which $X_t$ generates on $\Omega$ we have to conisder all pre-images of $\mathscr B(\mathbb R)$ w.r.t. $X_t$. Clearly, $$ X_t^{-1}(B) = \begin{cases} \emptyset,& \text{ if }B = \emptyset, \\ \Omega,& \text{ if }\{0,1\}\subseteq B, \\ \{t\},& \text{ if }1\in B,0\notin B \\ \Omega\setminus \{t\},& \text{ if }0\in B,1\notin B. \end{cases} $$ If it is difficult for you to derive this result - please tell me, I will expand it.

As a result we have $\mathscr F_t = \{\emptyset, \{t\},\Omega\setminus \{t\},\Omega\}$. Note that $\mathscr F$ contains all the elements of any $\mathscr F_t$ and $\mathscr F$ is the smallest such $\sigma$-algebra by definition, so:

  1. $\Omega\in \mathscr F$,

  2. $\emptyset\in \mathscr F$,

  3. Singleton set $\{t\}\in \mathscr F$ for any $t\in \Omega.$

It is necessary for $\mathscr F$ to contain all countable unions of its elements, and complements of its elements. As a result $\mathscr A\subseteq \mathscr F$. Here $\mathscr A$ is a class of all countable subsets of $\Omega$ (since they are countable unions of singletons) and their complements. It's easy to check that $\mathscr A$ is $\sigma$-algebra and hence $\mathscr F = \mathscr A$.

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It is easy to see that for any $t\in [0,1]$, $$X_t^{-1}(A)=\begin{cases}[0,1] &\text{ if } 0,1\in A\\ [0,1]-\{t\} & \text{ if } 0 \in A, 1\notin A\\ \{t\} & \text{ if } 0 \notin A, 1\in A\\ \phi &\text{ if } 0 \notin A, 1\notin A \end{cases}$$

So what is the $\sigma$-algebra generated by $\{X_t, 0\le t\le 1\}$?

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