Let $I:=[a,b], a,b \in \mathbb{R}, a<b$. Define $B_{I}:=B(\mathbb{R})|_{I}$ i.e. the Boreal sets of the reals restricted to the interval. Let $f:I \rightarrow \mathbb{R}$ be a step function.
Show that $f:(I,B_{I}) \rightarrow (\mathbb{R}, B(\mathbb{R}))$ i.e. the function is $I$-$B(\mathbb{R})$ measurable.
Attempt: By definition we say that a function is $A$-$B$ measurable if $f^{-1}(A) \in A_{1}, \forall A \in A_{2}$ where $A_{1}$ and $A_{2}$ denote sigma algebras.
So I need to show that for any $A \in B(\mathbb{R}), f^{-1}(A) \in \ B_{I}$. Let $A \in B(\mathbb{R})$ then $f^{-1}(A)=\{x \in I: f(x) \in A\}$
How do I evaluate what this set is equal to?