A central problem motivated by Diophantine approximation is to determine the size properties of subsets of of the form
where ‖⋅‖ denotes an arbitrary norm, I a denumerable set, (xi,ri)i∈ I a family of elements of × (0, ∞) and ϕ a nonnegative nondecreasing function defined on [0, ∞). We show that if FId, where Id denotes the identity function, has full Lebesgue measure in a given nonempty open subset V of , the set Fϕ belongs to a class Gh(V) of sets with large intersection in V with respect to a given gauge function h. We establish that this class is closed under countable intersections and that each of its members has infinite Hausdorff g-measure for every gauge function g which increases faster than h near zero. In particular, this yields a sufficient condition on a gauge function g such that a given countable intersection of sets of the form Fϕ has infinite Hausdorff g-measure. In addition, we supply several applications of our results to Diophantine approximation. For any nonincreasing sequence ψ of positive real numbers converging to zero, we investigate the size and large intersection properties of the sets of all points that are ψ-approximable by rationals, by rationals with restricted numerator and denominator and by real algebraic numbers. This enables us to refine the analogs of Jarník's theorem for these sets. We also study the approximation of zero by values of integer polynomials and deduce several new results concerning Mahler's and Koksma's classifications of real transcendental numbers.