The first exercise in my Measure and Integration book is to find the pre-image sigma algebra of various functions, namely $f(x)=x^3$, $x^2$ and $[x]$. You can correct me if I'm wrong but I'm fairly sure that for $x^3$ we have the power set for $\mathbb{R}$ and for $x^2$ we have the power set for $\mathbb{R^+}$, however for $[x]$ I'm quite unsure.
I don't ever really work on the greatest integer function however it's inverse must simply be for $f(x)=c$ the group all the real numbers greater than $c$ but less than $c+1$. How could I translate this into the preimage sigma algebra? My best guess on how to class these up would be the set $\left\{x+\frac{1}{n+\epsilon}: x\in\mathbb{R}\right\}, n\in\mathbb{N}$ however I'm very apprehensive that this could be wrong as I'm fairly new to measure theory.
In general if $f:X\to Y$ is a function and $\mathcal A$ is a $\sigma$-algebra on $Y$ then it is not difficult to prove that the collection: $$f^{-1}(\mathcal A):=\{f^{-1}(A)\mid A\in\mathcal A\}$$ is a $\sigma$-algebra on $X$.
You could call it the preimage $\sigma$-algebra of $\mathcal A$ with respect to $f$.
Let me note for completeness that here the notation $f^{-1}(A)$ stands for the set $\{x\in X\mid f(x)\in A\}$ and is not related with an eventual inverse of function $f$.
If $X=\mathbb R=Y$ and $f:\mathbb R\to\mathbb R$ is the greatest integer function and the codomain is equipped with $\sigma$-algebra $\mathcal P(\mathbb R)$ then for $A\in\mathcal P(\mathbb R)$ we find:$$f^{-1}(A)=\{x\in\mathbb R\mid\lfloor x\rfloor\in A\}=\bigcup_{n\in A\cap\mathbb Z}[n,n+1)$$
Notice that the sets of form $[n,n+1)$ with $n\in\mathbb Z$ form a partition of $\mathbb R$ and that the collection of unions of these intervals form indeed a $\sigma$-algebra.
We can describe this preimage $\sigma$-algebra as the collection:$$\left\{ \bigcup_{n\in I}\left[n,n+1\right)\mid I\subseteq\mathbb{Z}\right\} $$