Two definitions of Measurable Functions Let $\mathfrak M_n$ the Lesbegue-Measurable $\sigma$-algebra on ${\mathbb {R}}^n$. Let $f: E \longrightarrow \mathbb R$ with $E \in \mathfrak M_n$.


f is said to be measurable in the "first form" if $\forall A $ open subset of $\mathbb{R}$ it implies $f^{-1}(A) \in \mathfrak M_n$.
f is said to be measurable in the "second form" if $\forall A \in \mathfrak M_1$ it implies $f^{-1}(A) \in \mathfrak M_n$.


Is $ 1 \Rightarrow 2$ true?
There exists a continuous fuction $f$ and a Lebesgue measurable set $E$ such that $f^{-1} (E)$ is not measurable. Thus such a function $f$ satsfies the condition $1$ but not satisfies the condition $2.$
There exists a continuous fuction $f$ and a Lebesgue measurable set $E$ such that $f^{-1} (E)$ is not measurable. Thus such a function $f$ satsfies the condition $1$ but not satisfies the condition $2.$