Partial Differential Equations in Action

nand complementation. Precisely: Definition B.1 A collection Fof subsets of Ωis called σ−algebra if:(i) ∅↪ Ω∈ F;(ii) A ∈ Fimplies Ω∩A ∈ F;(iii) if {Ak}k∈N⊂ Fthen also ∪Ak and ∩Ak belong to F. Example B.1. If Ω= Rn, we can de show annotation

d/or intersections of open sets. ==Definition B.2 Given a σ−algebra Fin a set Ω, a measure on Fis a functionμ : F→ Rsuch that:(i) μ (A) ≥ 0 for every A ∈ F;(ii) if A1↪ A 2↪ … are pairwise disjoint sets in F↪ thenμ (∪k≥1Ak) = }k≥1μ (Ak) ( σ−additivity).The elements of Fare called measurable sets== .The Lebesgue measure in Rn is d show annotation