Measurable space

In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured.

It captures and generalises intuitive notions such as length, area, and volume with a set $$X$$ of 'points' in the space, but regions of the space are the elements of the σ-algebra, since the intuitive measures are not usually defined for points. The algebra also captures the relationships that might be expected of regions: that a region can be defined as an intersection of other regions, a union of other regions, or the space with the exception of another region.

Definition
Consider a set $$X$$ and a σ-algebra $$\mathcal F$$ on $$X.$$ Then the tuple $$(X, \mathcal F)$$ is called a measurable space.

Note that in contrast to a measure space, no measure is needed for a measurable space.

Example
Look at the set: $$X = \{1,2,3\}.$$ One possible $$\sigma$$-algebra would be: $$\mathcal {F}_1 = \{X, \varnothing\}.$$ Then $$\left(X, \mathcal{F}_1 \right)$$ is a measurable space. Another possible $$\sigma$$-algebra would be the power set on $$X$$: $$\mathcal{F}_2 = \mathcal P(X).$$ With this, a second measurable space on the set $$X$$ is given by $$\left(X, \mathcal F_2\right).$$

Common measurable spaces
If $$X$$ is finite or countably infinite, the $$\sigma$$-algebra is most often the power set on $$X,$$ so $$\mathcal{F} = \mathcal P(X).$$ This leads to the measurable space $$(X, \mathcal P(X)).$$

If $$X$$ is a topological space, the $$\sigma$$-algebra is most commonly the Borel $\sigma$-algebra $$\mathcal B,$$ so $$\mathcal{F} = \mathcal B(X).$$ This leads to the measurable space $$(X, \mathcal B(X))$$ that is common for all topological spaces such as the real numbers $$\R.$$

Ambiguity with Borel spaces
The term Borel space is used for different types of measurable spaces. It can refer to
 * any measurable space, so it is a synonym for a measurable space as defined above
 * a measurable space that is Borel isomorphic to a measurable subset of the real numbers (again with the Borel $$\sigma$$-algebra)