Various Operations

This chapter defines the image of local Lebesgue space, L 1 loc (Ω;E), measure under a semi-norm of E, where E is a Neumann space, and explains its properties. It considers the powers of a locally p -integrable real-valued measure. The chapter also defines finite piecewise constant measures and piecewise constant approximations. It states that E is sequentially separable if it contains a countable sequentially dense subset V. The chapter further presents some results on the separability of Lebesgue space. It shows that if E is a Frechet space, then Lebesgue space is also a Frechet space so, in this case, separability is equivalent to sequential separability. Finally, the chapter explains that the weak limit of a bounded sequence of measures in Lebesgue space itself belongs to Lebesgue space, provided it is locally integrable.