Spaceability of sets in Lp × Lq and C0 × C0

A subset E of an infinitely dimensional linearly-topological space X is called spaceable if there is an infinitely dimensional closed subspace Y of X   with Y⊂E∪{0}Y⊂E∪{0}. The main aim of the paper is to show the spaceability of the following sets:1.the set of those (f,g)∈Lp×Lq(f,g)∈Lp×Lq for which fg∉Lrfg∉Lr provided that one of the following conditions holds:(a)0<1p+1q<1r and sup⁡{μ(A):μ(A)<∞}=∞sup⁡{μ(A):μ(A)<∞}=∞;(b)1p+1q>1r and inf⁡{μ(A):μ(A)>0}=0inf⁡{μ(A):μ(A)>0}=0;2.the set of those (f,g)∈C0×C0(f,g)∈C0×C0 for which fg   is not integrable, where C0C0 is the space of continuous mappings which vanish at infinity;3.the set of those (f,g)∈Lp(G)×Lq(G)(f,g)∈Lp(G)×Lq(G) for which the convolution f⋆gf⋆g is not well-defined or is equal to ∞ provided G   is a locally compact but non-compact topological group and p,q>1p,q>1 with 1/p+1/q<11/p+1/q<1. The paper can be considered as a continuation of our previous ones in which we studied these sets from the Baire category and σ-porosity points of view.