On the essential unboundedness in measure of sequences of superlinear operators in classes Lϕ(L)Lϕ(L)

We establish a general theorem for a wide class of sequences of superlinear operators {Tn:L1(I2)→L0(I2),n=1,2,…} about existence of a function gg from a certain class Lϕ(L)Lϕ(L) such that the sequence of functions {Tn(g),n=1,2,…}{Tn(g),n=1,2,…} is essentially unbounded in measure on I2I2. This theorem implies several results about divergence of sequences of classical operators.