For positive integer r, let and be measure spaces.

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

Thus, this definition requires that "P" be a finite measure space.

He has also written highly cited papers on metric measure spaces.

This can be generalized to more than two σ-finite measure spaces.

In the setting of a locally compact group acting on a measure space there is a more general definition.

One may extend the notion of discrete measures to more general measure spaces.

While this approach does define a measure space, it has a flaw.

That said, the identities continue to hold, when properly formulated on a measure space.

Let be an open subset of , and be a measure space.