Large structures made of nowhere Lq functions

We say that a real-valued function f defined on a positive Borel measure space (X, mu) is nowhere q-integrable if, for each nonvoid open subset U of X, the restriction flu is not in L-q(U). When (X, mu) has some natural properties, we show that certain sets of functions defined in X which are p-integrable for some p's but nowhere q-integrable for some other q's (0 < p, q < infinity) admit a variety of large linear and algebraic structures within them. The presented results answer a question of Bernal-Gonzalez, improve and complement recent spaceability and algebrability results of several authors and motivate new research directions in the field of spaceability.