Lineability in subsets of measure and function spaces

We show, among other results, that if λ denotes the Lebesgue measure on the Borel sets in [0,1] and X is an infinite dimensional Banach space, then the set of measures whose range is neither closed nor convex is lineable in ca(λ,X). We also show that, in certain situations, we have lineability of the set of X-valued and non-σ-finite measures with relatively compact range. The lineability of sets of the type Lp(I)⧹Lq(I) is studied and some open questions are proposed. Some classical techniques together with the converse of the Lyapunov Convexity Theorem are used.