I would like to know how to find the σ-field generated by a random variable Y defined as Y(a)=Y(c)=1, Y(b)=Y(d)=-1 , under the sample space Ω = {a,b,c,d} and the probability measure P defined as P({a})=P({b})=1/4, P({c})=1/6 and P({d})=1/3.
I believe that the σ-field generated by a r.v. Y, to keep the same notation, is equal to Y-1[B(ℜ)] where B(ℜ) denotes the Borel σ-field.
What I'm thinking here is that the σ-field generated by Y will be {∅,Ω,{b,d},{a,c}}.
\begin{align} \sigma(Y) &= \sigma(\{Y^{-1}(B):B\in\mathcal B_{\mathbb R}\})\\ &= \sigma(\{Y^{-1}(\{1\}),Y^{-1}(\{-1\})\}\\ &= \sigma(\{\{a,c\},\{b,d\})\\ &= \{\varnothing, \{a,c\}, \{b,d\}, \Omega\}. \end{align}