Let $M$ be a smooth Riemannian manifold of dimension $d$. I wish to choose in a measurable way a map $C\_x:T\_xM\rightarrow \mathbb{R}^d$ s.t.
$$\forall u,v\in T\_xM: \langle C\_xu,C\_xv\rangle=\langle u,v\rangle\_x'$$
where $\langle\cdot , \cdot\rangle\_x'$ is some inner-product on the tangent space, which is defined as a function of the Riemannian metric inner-product over some measurable subset of $M$. I wish to choose a measurable $C\_x$ in its dependence on $x$, regarding the Borel measure space $M$.
I've been struggling and stuck with this for a while, and any hint would be appreciated.



A measurable choice of inner-product preserving linear maps between two vector spaces
Use a selection theorem to choose measurably a unit vector in $V\_x$. Consider now a new measurable collection of vector spaces $V'\_x$ given as the orthogonal complements of the chosen vectors. Iterate $d$-times. By now you got a measurable choice of orthogonal basis.
Fix an orthonormal basis in $W$. Use these bases to define $T\_x$.


