GSheaf

1. Preliminaries

Definition 1.1.

A $\sigma$ -algebra over a set $X$ is a subset $\Sigma\subseteq 2^X$ (that is, a set of subsets of $X$), satisfying:

1. If $I$ is some countable set ($0\leq|I|\leq\omega$), and $U_i\in\Sigma$ for $i\in I$, then $\bigcup_{i\in I}U_i \in \Sigma$.
2. If $U\in\Sigma$, then $X\setminus U\in\Sigma$.

A measurable space is a pair $(X, \Sigma)$, where $X$ is some set, and $\Sigma$ is a $\sigma$ -algebra over $X$.

In such a situation, a subset $S\subseteq X$ is called measurable iff $S\in\Sigma$.