Let $\mu$ be a compactly supported probability measure defined by a density function $f$ with respect to the Lebesgue measure and let $T$ be the operator given by the kernel $k(x,y)=f(y-x)$ (i.e. $T$ acts by convolution with $f$). For $B_n=$ the ball of radius $n$ in $\mathbb{R}^3$ the operator satisfies $T\chi_{B_n}-\chi_{B_n}\to 0$ in norm. $T$ has norm 1.
$I-T$ is injective, however not invertible.