This map extends in a canonical way to all bounded complex-valued measurable functions on " X ", and we have the following.
42.
This is a kind of Riesz representation theorem which allows for a measure to be represented as a linear functional on measurable functions.
43.
The same result holds if " f " and " g " are only assumed to be nonnegative measurable functions, by Tonelli's theorem.
44.
For a wide class of functions " f " ( namely : all continuous functions; all locally integrable functions; all non-negative measurable functions ).
45.
His lectures from 1902 to 1903 were collected into a " theory of measure and measurable functions and the analytical and geometrical definitions of the integral.
46.
In probability theory, a random variable is a measurable function from a probability space to a measurable space of values that the variable can take on.
47.
The space of measurable functions on a \ sigma-finite measure space ( X, \ mu ) is the canonical example of a commutative von Neumann algebra.
48.
Then there is a measure space and a real-valued essentially bounded measurable function " f " on " X " and a unitary operator such that
49.
We say that a set H of measurable functions satisfies the " separation property " if any two distinct functions in H belong to distinct equivalence classes.
50.
The monotone class theorem for functions can be a powerful tool that allows statements about particularly simple classes of functions to be generalized to arbitrary bounded and measurable functions.
