In this paper

we discuss the relationship between the property H for convergence in measure and the △2-condition for Orlicz function spaces Lφ equipped with the p-Amemiya norm ‖·‖Φ

p（1〈p〈∞）. From our results it follows that if Orlicz Function φ vanishes only at zero

then the Orlicz function spaces equipped with the p-Amemiya norm Lφ

phave property H

if and only if Orlicz function φ satisfies the △2-condition for all real numbers and φ is finitely valued; if φ vanishes outside zero

then Lφ

p have property Hμ if and only if φ satisfies the △2- condition at infinity and φ is finitely valued. Analogous results are also proved for the subspace Eφ

pof order contin- uous elements of the Orlicz function space Lφ

p.