Appendix A: Mathematical Complements
A.1 Functions and Integration Analog signals are modeled by measurable functions. We first give the main theorems of Lebesgue integration. A function f is said to be integrable if ∫ +∞ −∞ |f(t)| dt < +∞. The space of integrable functions is written L(R). Two functions f1 and f2 are equal in L(R) if ∫ +∞ −∞ |f1(t)− f2(t)| dt = 0. This means that f1(t) and f2(t) can differ only on a set of points of measure 0. We say that they are almost everywhere equal. The Fatou lemma gives an inequality when taking a limit under the Lebesgue integral of positive functions. Lemma A.1 (Fatou). Let {fn}n∈N be a family of positive functions fn(t) ! 0. If limn→+∞ fn(t) = f(t) almost everywhere then ∫ +∞ −∞ f(t) dt " lim n→+∞ ∫ +∞ −∞ fn(t) dt.