Some properties of functions in Orlicz space

For functions in Orlicz space $L^*_M$, we study the behavior of $\int^\tau_0x^*(t)\,dt$, where $x^*(t)$ is non-increasing and equimeasurable with $|x(t)|$. We establish the existence of unbounded functions in $L^*_M$, that are not limits of bounded functions and for which $\int_0^\tau x^*(t)\,dt=o(\tau M^{-1}(1/\tau))$. Moreover, we establish a new criterion for an $N$-function to belong to the class $\Delta_2$ and a sufficiency test for a function to belong to Orlicz space.