I'm trying to do exercises $2.N.$, $2.O.$, and $2.P.$ from Chapter $2$ of Bartle's The Elements of Integration and Lebesgue Measure.
Here X is the $\sigma$-algebra in $X$.
I already did the exercise $2.N$. Now I'm trying to do the exercises $2.O.$ and $2.P.$. But I'm not understanding a passage in exercise $2.O.$:
Let $A$ ( or A?) be a collection of subsets of $Y$ such that $f^{-1}(E)\in$ X for every $E\in A$ (or $E\in$ A??).
The notation for $\sigma$-algebra that Bartle has adopted causes a lot of confusion.
I think it is important to solve exercise $2.O.$ because I think I will use it to solve the exercise $2.P.$.
To avoid confusion I denote collections of sets by curly letters.
By 2.N, $\mathcal{Y}=\{E\subset Y:f^{-1}(E)\in \mathcal{X}\}$ is a $\sigma$-algebra, which contains $\mathcal{A}$ (because $f^{-1}(E)\in\mathcal{X}$ for all $E\in \mathcal{A}$). Therefore, it contains the $\sigma$-algebra generated by $\mathcal{A}$ (which is the smallest $\sigma$-algebra containing $\mathcal{A}$).
According to Defintion 2.3 on page 8, for 2.P, take $\mathcal{A}=\{(\alpha,\infty):\alpha\in \mathbb{R}\}$ and note that $\mathcal{A}\subset\mathcal{B}(\mathbb{R})$ and $\sigma(\mathcal{A})=\mathcal{B}(\mathbb{R})$.