Notes for “Wavelets and Computation”

• The exterior measure of any set A ⊂ R is μ∗(A) = inf E⊂A μ(E) where the infimum is taken over all elementary sets. • A is Lebesgue measurable if for all > 0 there exists an open set O ⊃ A such that μ∗(O \A) ≤ . • The (Lebesgue) measure of a measurable set is μ(A) = μ∗(A). A function f is measurable if the sets {t | f(t) ≤ a} are measurable for all a ∈ R. Definition 2 (Lebesgue integral) • For a simple function s(t) = ∑n i=1 ciχAi(t) the Lebesgue integral is ∫