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 $R:={\mathcal {P}}(U)$ , the power set of $U$ .

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