<P> In mathematics and in particular measure theory, a measurable function is a function between two measurable spaces such that the preimage of any measurable set is measurable, analogously to the definition that a function between topological spaces is continuous if the preimage of each open set is open . In real analysis, measurable functions are used in the definition of the Lebesgue integral . In probability theory, a measurable function on a probability space is known as a random variable . </P> <P> Let (X, Σ) (\ displaystyle (X, \ Sigma)) and (Y, T) (\ displaystyle (Y, \ mathrm (T))) be measurable spaces, meaning that X (\ displaystyle X) and Y (\ displaystyle Y) are sets equipped with respective σ (\ displaystyle \ sigma) - algebras Σ (\ displaystyle \ Sigma) and T (\ displaystyle \ mathrm (T)). A function f: X → Y (\ displaystyle f: X \ to Y) is said to be measurable if the preimage of E (\ displaystyle E) under f (\ displaystyle f) is in Σ (\ displaystyle \ Sigma) for every E ∈ T (\ displaystyle E \ in \ mathrm (T)); i.e. </P>

When is a function said to be measurable