Lecture 1 Measurable Spaces Families of Sets

Families of Sets Definition 1.1 (Order properties). A countable1 family {An}n∈N of 1 To make sure we are all on the same page, let us fix some notation and terminology: • ⊆ denotes a subset (not necessarily proper). • A set A is said to be countable if there exists an injection (one-to-one mapping) from A into N. Note that finite sets are also countable. Sets which are not countable are called uncountable. • For two functions f : B→ C, g : A→ B, the composition f ◦ g : A → C of f and g is given by ( f ◦ g)(x) = f (g(x)), for all x ∈ A. • {An}n∈N denotes a sequence. More generally, (Aγ)γ∈Γ denotes a collection indexed by the set Γ. • We use the notation An ↗ A to denote that the sequence {An}n∈N is increasing and A = ∪n An. Similarly, An ↘ A means that {An}n∈N is decreasing and A = ∩n An. subsets of a non-empty set S is said to be