Measurable space

Remarks

The real line with its Borel σ-algebra $\mathcal{B}(\R)$ is a measurable space, because $\mathcal{B}(\R)$ is a σ-algebra by its definition. Any set $X$ with its power set is a measurable space, and so is $X$ with $\set{\varnothing, X}$. For $X = \set{1, 2}$, the pair $(X, \set{\varnothing, \set{1}, X})$ is not a measurable space. The family is not closed under complements, since $X \setminus \set{1} = \set{2}$ is missing. A map $f \colon X \to Y$ between measurable spaces $(X, \Sigma)$ and $(Y, T)$ is called measurable when $f^{-1}(B) \in \Sigma$ for every $B \in T$. Some authors call a measurable space a Borel space.