Approximate point spectrum of multiplication operator in $L^p$

Question

Let $M_\phi f=\phi f, \,\, \phi\in L^\infty(X,\Omega,\mu),\,\, f\in L^p(X,\Omega,\mu),\, 1\leqslant p \leqslant \infty.$

I finded $$\sigma(M_\phi)=\left\{ \lambda \in \mathbb{C} \mid \not \exists \epsilon>0 \colon |\lambda-f(x)|\geqslant \epsilon \,\,\text{a.e.} \right\}$$ and $$\sigma_p(M_\phi)=\{ \lambda \in \mathbb{C} \colon \mu\left( \{x\in X \colon f(x)=\lambda \} \right)>0 \}.$$

$\sigma_{ap}(M_\phi)=\,?$

Answer 1

$M_\phi$ is normal $\Rightarrow \sigma_{ap}=\sigma$

Answer 2

$$ \sigma_{ap}(M_\phi) = \{ \lambda\in\mathbb{C} : \mu\{ \nu : 0 <|\phi(\nu)-\lambda| < \epsilon\} >0\;\; \forall \epsilon > 0 \}. $$