If $(X, \mathcal{A})$ and $(Y, \mathcal{B})$ are measurable spaces and a function $f : (X, \mathcal{A}) \hookrightarrow (Y, \mathcal{B})$ . We say that $f$ is $(\mathcal{A}, \mathcal{B})$-measurable if $\forall E \in \mathcal{B} : f^{-1}(E) \in \mathcal{A}$ .
Let $\mathcal{L}(\Bbb{R}^n)$ denote the Lebesgue $\sigma$-algebra of $\Bbb{R}^n$
In order to use this to solve an excercise, I want to know if the canonical projections of $\Bbb{R}^n$
$$\forall k \in [1,n] \cap \Bbb{Z}, \pi_k : \Bbb{R}^n \hookrightarrow \Bbb{R}, (x_j)_{j=1}^n \mapsto x_k$$
are $(\mathcal{L}(\Bbb{R}^n), \mathcal{L}(\Bbb{R}))$-measurable. I have tried to think about it but I am not getting anywhere.
In my book the only there is a characterization about real valued measurable functions which says that $f: X \hookrightarrow \Bbb{R}$ is measurable iff $\forall t \in \Bbb{R} : f^{-1}(-\infty, t) $ is measurable on $X$.
The problem is that I don't know if in this characterization the $\sigma$-algebra considered on the codomine $\Bbb{R}$ is the Borel or the Lebesgue and the book does not say anything about that.
Thanks in advance :)