measurable space
Measurable space
A measurable space is a pair \[(X,\mathcal{A})\] consisting of a non-empty set \[X\] and a sigma-algebra, which defines the subsets that will be measured.
Definition
Consider a non-empty set \[X\] and a sigma-algebra \[\mathcal{F}\] on \[X\]. Then the tuple \[(X,\mathcal{F})\] is called a measurable space. The elements of \[\mathcal{F}\] are called measurable sets within the measurable space.
Assume we have set \[X=\left\{ 1,2,3 \right\}\]. One possible sigma-algebra would be \[\mathcal{F}_{1}=\left\{ X,\emptyset \right\}\]. Then, \[(X,\mathcal{F}_{1})\] is a valid measurable space.