Moreover, if $\psi$ satisfies that $A_\psi(\alpha)\neq 1$ for some $\alpha>1$ (see Equation ), there are holomorphic functions $f\in \cap_{q>p} B^{1/q}_{q,q}(S^1)$ such that $|H_{\bar{f}}|^p\in \mathcal{M}_\psi$ is non-measurable.

Since we only consider $p$:th powers of operators, our results extend mutatis mutandis to $0<p<1$. We restrict our attention to $p\geq 1$ in order to avoid quasi-normed Banach spaces.

In Section \[prelsection\] we provide an overview of the theory of Lorentz ideals from an extrapolation point of view. The general form of Theorem \[mainthmintro\] will be considered in Section \[hanekopdldl\]. We consider the special case $p=2,4,6$ of Theorem \[mainthmintro\] in Section \[p246sec\] where a result of Janson-Upmeier-Wallsten allows us to give exact formulas for the Dixmier trace $\mathrm{Tr}_{\omega,\psi}(|H_{\bar{f}}|^p)$. Finally, in Section \[Non-meas\] we construct holomorphic functions $f\in \cap_{q>p} B^{1/q}_{q,q}(S^1)$ such that $|H_{\bar{f}}|^p\in \mathcal{M}_\psi$ is non-measurable.\
[**Acknowledgements:**]{} We are grateful to Genkai Zhang for interesting discussions and particularly for introducing the work [@JUW] to us. We also thank Fedor Sukochev and Evgeniy Semenov for helpful comments on the distance formula to the separable part of a Lorentz ideal. We also acknowledge support from the Swedish Research Council Grant 2015-00137 and Marie Sklodowska Curie Actions, Cofund, Project INCA 600398. This work was finalized at the Erwin 