Measurable Space

A measurable space is a foundational concept in measure theory. It provides the structure needed to define and work with measures, which are generalizations of intuitive notions such as length, area, and volume.

Definition:

A measurable space is an ordered pair $(X, \mathcal{F})$ where:

1. $X$ is a set.
2. $\mathcal{F}$ is a σ-algebra on $X$.

The set $X$ is often referred to as the sample space or universal set, while the σ-algebra $\mathcal{F}$ is a collection of subsets of $X$ with specific properties that make them "measurable".

σ-Algebra:

A collection $\mathcal{F}$ of subsets of $X$ is called a σ-algebra if it satisfies the following three properties:

1. $X$ itself is in $\mathcal{F}$ (i.e., the entire space is measurable).
2. If $A$ is in $\mathcal{F}$, then its complement $A^c$ is also in $\mathcal{F}$.
3. If $A_1, A_2, ... \in \mathcal{F}$, then the countable union $\cup_{i=1}^{\infty}A_i$ is also in $\mathcal{F}$.

Explanation:

The idea of a σ-algebra is to delineate which subsets of $X$ are "measurable". Not every subset of $X$ necessarily needs to be in $\mathcal{F}$, but $\mathcal{F}$ should be closed under complementation and countable unions (and, by extension, countable intersections, since the complement of a union is the intersection of the complements).

Once you have a measurable space $(X, \mathcal{F})$, you can then define a measure on this space, turning it into a measure space. The measure assigns to each measurable set (i.e., each set in the σ-algebra) a non-negative real number (or +∞) in a consistent and additive manner. This additive consistency is what lets us work with integrals and probabilities in a rigorous manner.