Measurable space

Consider a set ${\displaystyle X}$ and a σ-algebra ${\displaystyle {\mathcal {F}}}$ on ${\displaystyle X.}$ Then the tuple ${\displaystyle (X,{\mathcal {F}})}$ is called a measurable space.^[2] The elements of ${\displaystyle {\mathcal {F}}}$ are called measurable sets within the measurable space.

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

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

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

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