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 $N_F$ be the Nemytskiĭ set-valued operator induced by a sup-measurable set-valued function $F:Ω × X → 2^{Y}$. It is shown that if $N_F$ maps a modular space $(N(L(Ω,Σ,μ;X)), ϱ_{N,μ})$ into subsets of a modular space $(M(L(Ω,Σ,μ;Y)),ϱ_{M,μ})$, then $N_F$ is automatically modular bounded, i.e. for each set K ⊂ N(L(Ω,Σ,μ;X)) such that $r_K = sup{ϱ_{N,μ}(x) : x ∈ K} < ∞$ we have $sup{ϱ_{M,μ}(y): y ∈ N_F(K)} < ∞$.