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Let $S(0,1)$ be the $*$-algebra of all classes of Lebesgue measurablefunctions on the unit interval $(0,1)$ and let $(\mathcal{A},\left\|\cdot\right\|_\mathcal{A})$ be a complete symmetric $\Delta$-normed $*$-subalgebraof $S(0,1)$, in which simple functions are dense, e.g., $L_\infty (0,1)$,$L_{\log}(0,1)$, $S(0,1)$ and the Arens algebra $L^\omega (0,1)$ equipped withtheir natural $\Delta$-norms. We show that there exists no non-trivialderivation $ \delta: \mathcal{A} \to S(0,1)$ commuting with all dyadictranslations of the unit interval. Let $\mathcal{M}$ be a type $II$ (or$I_\infty$) von Neumann algebra, $\mathcal{A}$ be its abelian von Neumannsubalgebra, let $S(\mathcal{M})$ be the algebra of all measurable operatorsaffiliated with $\mathcal{M}$.  We show that any non-trivial derivation $\delta:\mathcal{A} \toS(\mathcal{A})$ can not be extended to a derivation on $S(\mathcal{M})$. Inparticular, we answer an untreated question in \cite{BKS1}.

Non-existence of translation-invariant derivations on algebras of measurable functions