Bochner measurable function


In mathematics - specifically, in functional analysis - a Bochner-measurable function taking values in a Banach space is a function that equals a.e. the limit of a sequence of measurable countably-valued functions, i.e.,
where the functions each have a countable range and for which the pre-image is measurable for each x. The concept is named after Salomon Bochner.
Bochner-measurable functions are sometimes called strongly measurable, -measurable or just measurable.

Properties

The relationship between measurability and weak measurability is given by the following result, known as Pettis' theorem or Pettis measurability theorem.

Function f is almost surely separably valued if there exists a subset N ⊆ X with μ = 0 such that f ⊆ B is separable.


A function f : X → B defined on a measure space and taking values in a Banach space B is measurable if and only if it is both weakly measurable and almost surely separably valued.

In the case that B is separable, since any subset of a separable Banach space is itself separable, one can take N above to be empty, and it follows that the notions of weak and strong measurability agree when B is separable.