A landau-type construction concerning (Lp)′ ⊂ Lq

Let 1 ⩽ p < ∞ and 1/p + 1/q = 1. For a locally finite measure space (X, S, μ) and a measurable complex-valued function f ∉ Lq functions g ∈ Lp may be constructed explicitly which satisfy∫x|f(x)g(x)|dμ(x)=∞