Let $X$ be a separable Banach space and $\mathcal L$ the collection of bounded linear operators on $X$. The strong operator topology has the sub-basis $\{B_{x,y,\epsilon}\colon x,y\in X,\epsilon>0\}$, where $B_{x,y,\epsilon}=\{T\colon \|Tx-y\|<\epsilon\}$. The Borel $\sigma$-algebra generated by this topology is called the strongly measurable $\sigma$-algebra on $\mathcal L$.

Now the question (which has arisen in a study of the multiplicative ergodic theorem on Banach spaces):

Let $K=\{T\in\mathcal L\colon \text{ker}(T)\ne\{0\}\}$. Is this set strongly measurable?

Thanks for any information.