Indeed, let be a non-negative measurable function defined over the measure space as before.

Let a measurable function be injective and generating, then the following two conditions are equivalent:

"Perfect" means that for every measurable function from to the image measure is regular.

As T is a measurable function, for all i, there exists such that .

The range of a lifting is always a set of measurable functions with the "separation property".

Theorem 2: Let be independent uncertain variables, and measurable functions.

Theorem: Let be independent uncertain variables, and a measurable function.

Let be a measurable space and let be an -measurable function.

A theory of measurable functions and integrals on these functions.

We extend the integral by linearity to non-negative measurable simple functions.