AbstractThis thesis deals with a certain set function called entropy and its ápplications to some problems in classical Fourier analysis. For a set S ⊆ [0, 1e] the entropy of S is defined by E(S) = infS⊂∪kIk,Ikintervals Σk | Ik | log(1|Ik|). We begin by using notions related to entropy in order to investigate the maximal operator MΩ given by MΩ(f)(x) = supr>0(1rn) ∫|t| ≤r Ω(t) |f(x + t)| dt, f ϵ L1(Rn), where Ω is a positive function, homogeneous of degree 0, and satisfying a certain weak smoothness condition. Then the set function entropy is investigated, and certain of its properties are derived. We then apply these to solve various problems in differentiation theory and the theory of singular integrals, deriving in the process, entropic versions of the theorems of Hardy and Littlewood and Calderón and Zygmund.