1. For finite measures and, the idea is to consider functions with. 2. We wish to obtain a generally finite measure as the bin size goes to zero. 3. More generally the spaces with an atomless, finite measure and are not locally convex. 4. This property is false without the assumption that at least one of the has finite measure . 5. It turns out that | & mu; | is a non-negative finite measure . 6. If is a finite measure on, the function admits for the convergence in measure the following fundamental system of neighborhoods 7. The fact that the remaining part of is singular with respect to follows from a technical fact about finite measures . 8. Conversely, any homogeneous system of imprimitivity is of this form, for some measure ?-finite measure ?. 9. For instance, it is used in proving the uniqueness claim of the Carath�odory extension theorem for ?-finite measures . 10. Once the result is established for finite measures , extending to-finite, signed, and complex measures can be done naturally.
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