ABSTRACT
The fundamental theorem of calculus presented above for Lebesgue measures can be generalized to arbitrary Radon measures. It turns out that the same approach works if a different covering theorem is employed instead of the Vitali theorem. This covering theorem is the Besicovitch covering theorem which we present in this section. It is necessary because for a general Radon measure, µ, it is no longer the case that the measure is translation invariant. This implies that there is no way to estimate μ ( B ^ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203711316/6d18f3d9-46dc-4024-b299-28e5cf12196b/content/inequ15_263_1.tif"/> in terms of µ(B) and thus the Vitali covering theorem is of no use. The following theorem is the Besicovitch covering theorem. Note that the balls in the covering are not enlarged as they are in the Vitali theorem.
