Let $L_1(\mathbb T)$ be considered as a closed ideal of $M(\mathbb T)$, the Banach algebra of measures on the circle. Then $M(\mathbb T)$ can be identified with the multiplier algebra of $L_1(\mathbb T)$ and so it carries the strict topology.

Is there a measure that cannot be approximated by invertible measures in the strict topology?

Put differently, what is the strict closure of invertible measures in $M(\mathbb T)$?