Sigma-Finite Measure

• categories: Measure Theory, Definition

Definition:
A measure $\mu$ defined on a measurable space $(X,\mathcal{F})$ is called $\sigma$-finite if $X$ can be expressed as a countable union of measurable sets $\{A_n{\}}_{n=1}^{\infty }$ such that:

1. $\mu (A_n)<\infty$ for each $n$.
2. $A_n\in \mathcal{F}$ for all $n$.

Formally:

$$X=\bigcup _{n=1}^{\infty }{A}_n\text{with}\mu (A_n)<\infty .$$

Intuition:
A measure $\mu$ is $\sigma$-finite if we can decompose the space $X$ into a sequence of smaller, finite measure pieces. This concept generalizes the idea of finite measures to larger spaces by allowing for countable coverings with finite measure components.

Examples:

• The Lebesgue Measure on $\mathbb{R}$ is $\sigma$-finite because $\mathbb{R}$ can be covered by intervals $(-n,n)$ for $n\in \mathbb{N}$, and each interval has finite Lebesgue measure.
• Counting measure on any countable set (e.g., $\mathbb{N}$) is $\sigma$-finite because we can decompose the set into singletons, each with measure 1.

Non-Example:

• A measure that assigns $\infty$ to the entire space and cannot be decomposed into finite measure parts is not $\sigma$-finite.