On automatic boundedness of Nemytskiĭ set-valued operators

Let X, Y be two separable F-spaces. Let (Ω,Σ,μ) be a measure space with μ complete, non-atomic and σ-finite. Let ${\displaystyle N_F}$ be the Nemytskiĭ set-valued operator induced by a sup-measurable set-valued function ${\displaystyle F:\Omega \times X\to 2^Y}$. It is shown that if ${\displaystyle N_F}$ maps a modular space ${\displaystyle (N\left(L(\Omega ,\Sigma ,\mu ;X)\right),{\varrho }_{N,\mu })}$ into subsets of a modular space ${\displaystyle (M\left(L(\Omega ,\Sigma ,\mu ;Y)\right),{\varrho }_{M,\mu })}$, then ${\displaystyle N_F}$ is automatically modular bounded, i.e. for each set K ⊂ N(L(Ω,Σ,μ;X)) such that ${\displaystyle r_K=\supset \{{\varrho }_{N,\mu }\left(x\right):x\in K\}<\infty }$ we have ${\displaystyle \supset \{{\varrho }_{M,\mu }\left(y\right):y\in N_F\left(K\right)\}<\infty }$.