Measurable space

Consider a set X and a σ-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. 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) ==See also==