Abstract :
We study the space L0(μ) of μ-measurable real or complex-valued functions which vanish outside a set of finite measure, equipped with the group-norm given by ||ƒ||0 = μ{ω|ƒ(ω) ≠ 0}. Attention is focussed on the invariance of || ||0 under homotheties. A class of generalized Köthe function spaces with homothetically invariant group-norms is examined, and the results are specialized to L0(μ). We show that although several properties of L0 are shared by these more general spaces, L0(μ) is essentially the only generalized Köthe function space with respect to μ whose group-norm is simulianeously invariant under homotheties and additive in the L1 sense, i.e., ||ƒ + g|| = ||ƒ|| + ||g|| whenever |ƒ| ∧ |g| = 0 in the usual (μ-a.e.) lattice order.
