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

From its definition, the electron density is a non-negative function integrating to the total number of electrons.

Here ' is Lebesgue measure and is a non-negative measurable function.

Define where is a non-negative symmetric function in and that can be chosen by the user.

Since both and f(x) are non-negative functions, it follows that .

Let be a non-negative increasing function on the half-line such that .

Let be a non-negative continuous function on the half-line and .

Note that both f and f are non-negative functions.

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

We have defined the integral of "f" for any non-negative extended real-valued measurable function on "E".