In the lemma below, is the -algebra of Borel sets on If and is a measurable space, then
is the smallest -algebra on such that is -measurable.
Let be a function, and a measurable space. A function is -measurable if and only if for some -measurable [1]
Remark. The "if" part simply states that the composition of two measurable functions is measurable. The "only if" part is proven below.
Proof. |
Let be -measurable.
Assume that is an indicator of some set If then the function suits the requirement. By linearity, the claim extends to any simple measurable function
Let be measurable but not necessarily simple. As explained in the article on simple functions, is a pointwise limit of a monotonically non-decreasing sequence of simple functions. The previous step guarantees that for some measurable The supremum exists on the entire and is measurable. (The article on measurable functions explains why supremum of a sequence of measurable functions is measurable). For every the sequence is non-decreasing, so which shows that |
Remark. The lemma remains valid if the space is replaced with where is bijective with and the bijection is measurable in both directions.
By definition, the measurability of means that for every Borel set Therefore and the lemma may be restated as follows.
Lemma. Let and is a measurable space. Then for some -measurable if and only if .