Let Omega be an open ball in R.

N and B the sigma-algebra of Borel setson Omega. The Lebesgue decomposition theorem states that if mu is a finite measure on (Omega, B), then there exists a finite measure mu_1 which is absolutely continuous with respect to the Lebesgue measure lambda.

N and a finite measuremu_2 which is singular with respect to lambda.

N such that mu = mu_1 + mu_2,and mu_1 and mu_2 are unique. Moreover, the Radon-Nikodym theorem states that there exists a function f in L.

1(Omega) such that mu_1(E) = int_E f dx. For a real number p greater than 1, we define a set function cap_p( . , Omega) on B that is countably subadditive. We present a proof of a statement very similar to the classical Lebesgue decomposition, namely an arbitrary finite measure mu on (Omega, B) is uniquely decomposed into a finite measure mu_1.

*which is absolutely continuous with respect to cap_p( . , Omega) and a finite measure mu_2 which is singular with respect to cap_p( . , Omega). We then present a proof of a statement very similar to the Radon-Nikodym theorem regarding mu_1.

*, namely mu_1.

* is decomposed into a finite signed measure mu_0 whichbelongs to L.

1(Omega) and a finite signed measure mu_1 which belongs to theSobolev dual space (W.

{1,p}(Omega)).

*, so that mu = mu_0 + mu_1 + mu_2.