In mathematics—specifically, in functional analysis—a weakly measurable function taking values in a Banach space is a function whose composition with any element of the dual space is a measurable function in the usual (strong) sense.
For separable spaces, the notions of weak and strong measurability agree.
is a Banach space over a field
(which is the real numbers
or complex numbers
is said to be weakly measurable if, for every continuous linear functional
is a measurable function with respect to
and the usual Borel
A measurable function on a probability space is usually referred to as a random variable (or random vector if it takes values in a vector space such as the Banach space
Thus, as a special case of the above definition, if
is a probability space, then a function
-valued) weak random variable (or weak random vector) if, for every continuous linear functional
-valued random variable (i.e. measurable function) in the usual sense, with respect to
and the usual Borel
The relationship between measurability and weak measurability is given by the following result, known as Pettis' theorem or Pettis measurability theorem.
is said to be almost surely separably valued (or essentially separably valued) if there exists a subset
Theorem (Pettis, 1938) — A function
defined on a measure space
and taking values in a Banach space
is (strongly) measurable (that equals a.e.
the limit of a sequence of measurable countably-valued functions) if and only if it is both weakly measurable and almost surely separably valued.
above to be empty, and it follows that the notions of weak and strong measurability agree when