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