# Measure Theory/Advanced set theory

## Structures of set theory

We begin by recalling the basic notions of set theory, which should be familiar to everybody:

Definition 1.1:

Let ${\displaystyle S,T\subseteq U}$  be subsets of some universal set ${\displaystyle U}$ .

• The union of ${\displaystyle S}$  and ${\displaystyle T}$  is ${\displaystyle S\cup T:=\{u\in U|u\in S\vee u\in T\}}$ .
• The intersection of ${\displaystyle S}$  and ${\displaystyle T}$  is ${\displaystyle S\cap T:=\{u\in U|u\in S\wedge u\in T\}}$ .
• The sum of ${\displaystyle S}$  and ${\displaystyle T}$  is ${\displaystyle S+T:=(S\cup T)\setminus (S\cap T)}$ .

We will follow the convention to denote the intersection of ${\displaystyle S}$  and ${\displaystyle T}$  just by juxtaposition: ${\displaystyle ST:=S\cap T}$ .

Given a universal set ${\displaystyle U}$ , the subsets of ${\displaystyle U}$  have the algebraic structure of a ring:

Theorem 1.2:

Let ${\displaystyle U}$  be a universal set. Let ${\displaystyle R:={\mathcal {P}}(U)}$ , the power set of ${\displaystyle U}$ .