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 $S,T\subseteq U$ be subsets of some universal set $U$.

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

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

Given a universal set $U$, the subsets of $U$ have the algebraic structure of a ring:

Theorem 1.2:

Let $U$ be a universal set. Let $R:=𝒫(U)$, the power set of $U$.