# Real and Complex analysis

## Measurability

**Definition**

- 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:
- $X \in \mathfrak M$
- If $A \in \mathfrak M$, then $A^c \in \mathfrak M$, where $A^c$ is the complement of $A$ relative to $X$.
- If $A \in \bigcup_{n=1}^{\infty} A_n$ and if $A_n \in \mathfrak M$ for $n$ =1, 2, 3, ..., then $A\in\mathfrak M$.

- 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 $X$.
- 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 $Y$.