Let $A^+$ denote the Moore-Penrose inverse as a function on the space of real $n\times m$ matrices $A\in\mathbb{R}^{n\times m}$. Is this function $\mathcal B(\mathbb{R}^{n\times m})$ measurable?
From here we have the following limit relation: $A^+=\lim_{\delta\downarrow 0} (A'A+\delta I_m)^{-1}A'$, so in particular we have $A^+=\lim_{k\to \infty} (A'A+\frac{1}{k} I_m)^{-1}A'$. Matrix transpose and matrix multiplication are continuous on $\mathbb{R}^{n\times m}$, and matrix inversion on the set of invertible matrices $A\in\mathbb R^{m\times m}$ is also continuous (see here). Since continuous functions are measurable, we get that $A^+$ is the pointwise limit of a sequence of measurable functions, hence measurable.
Is this reasoning ok? Thank you for your help.
