What is the measure of this set of matrices

Question

I am interested in the following question about matrices:

Let $U\in\mathbb R^{n\times n}$ be considered as a map from $(\mathbb R^n,||\cdot||)$, a normed space, to itself. We say that $U$ is good if it satisfies $||(U+I)^{-1}||+||(I+U^{-1})^{-1}||\leq 1$ where $I$ is the identity matrix. Let $\mu$ be the Haar probability measure on $O_n(\mathbb R)$.

The norm we take on the matrices is the operator norm.

Then I have two questions:

1.What is the $\mu$ measure of the set of good matrices?

2.Can you characterize all good $U$'s?

Thanks.

Remark: for me, question 2 is somewhat more important, so feel free to answer the one and not the other.