Real and Complex analysis

Measurability

Definition

1. A collection $\mathfrak M$ of subsets of a set $X$ is said to be a $\sigma$-algebra in $X$ if $\mathfrak M$ has following properties:
   1. $X \in \mathfrak M$
   2. If $A \in \mathfrak M$, then $A^c \in \mathfrak M$, where $A^c$ is the complement of $A$ relative to $X$.
   3. If $A \in \bigcup_{n=1}^{\infty} A_n$ and if $A_n \in \mathfrak M$ for $n$ =1, 2, 3, ..., then $A\in\mathfrak M$.
2. If $\mathfrak M$ is $\sigma$-algebra in $X$, the $X$ is a measurable space. and the members of $\mathfrak M$ are called the measurable sets in $X$.
3. If $X$ is a measurable space, $Y$ is topological space, and $f:X \to Y$, the $f$ is said to be measurable provided that $f^{-1}(v)$ is a measurable set in $X$ for every open set $v$ in $Y$.