Let $M\subset \mathbb{R}^k$ compact. For every $x\in M$ we define $L(x): \mathbb{R}^m \rightarrow \mathbb{R}^m $ a linear isomorphism
Let $G_n (\mathbb{R}^m)=\{ W: W\ \mbox{is subspace of} \ \mathbb{R}^m \ \mbox{with} \dim(W)=n\}$ and fixed a topology $\tau$ in $G_n (\mathbb{R}^m)$.
Suppose that the correspondence $M \ni x \mapsto E(x) \in G_n (\mathbb{R}^m)$ is Borel-measurable and $x \mapsto L(x)$ is continuous. I wonder if the application $x \mapsto L(x)\vert E(x)$ is Borel-measurable?
Note: $L(x)\vert E(x)$ is restriction operator $L(x)\vert E(x): E(x)\rightarrow \mathbb{R}^m $
thanks for any suggestions
The question arises in trying to prove the measurability of Lyapunov exponents according to the book of Ricardo Mañe.