$(\Omega, \mathcal{F},\mathbb{P})$ and $A \in \mathcal{F}) = ((0,1], \mathcal{B}((0,1]), \lambda )$, where $\lambda$ is the Lebesgue measure. Define
$X_1(\omega)=0, \forall\omega\in\Omega$
$X_2(\omega)=\mathbf{1}_{1/2} (\omega)$
$X_3(\omega)=\mathbf{1}_{\mathbb{Q}}(\omega) $
where $\mathbb{Q}$ is the set of rational numbers in (0, 1]. Note that
$\mathbb{P}(X_1 = X_2 = X_3) = 1$
and find
$\sigma(X_i)$ for $i = [1, 2, 3]$
I have tried to solve the exercise using the Borel-Cantelli lemma and the formula of $\sigma$ but I have got some problems in the definition of the different values assumed.
$\sigma (X_i)$ is not the standard deviation of $X_i$, it is the sigma-field generated by the random variable $X_i$. That is,
$$\sigma (X_i) = \{ X_i^{-1}(B) : B \in \mathcal{B} \}$$
Where
$$X_i^{-1} (B) = \{ \omega \in \Omega: X_i(\omega) \in B \}$$
A class mate of yours asked this exact question earlier in the week. I tried to find it for reference but couldn't find it.
For example, for $X_1$ every $\omega$ is mapped to $0$, and so the preimage of every set $B$ is either $\emptyset$ if $0 \notin B$ or $\Omega$ if $0 \in B$. Hence
$$\sigma (X_1) = \{ \emptyset, \Omega \}$$
If this is confusing to you, I would recommend reading the first chapter of "Probability: Theory and Examples" by Durrett,