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Let $p>n$ and let $L^1_p(R^n)$ be a homogeneous Sobolev space. For anarbitrary Borel measure $\mu$ on $R^n$ we give a constructive characterizationof the space $L^1_p(R^n)+L_p(R^n;\mu)$. We express the norm in this space interms of certain oscillations with respect to the measure $\mu$. This enablesus to describe the $K$-functional for the couple $(L_p(R^n;\mu),L^1_p(R^n))$ interms of these oscillations, and to prove that this couple isquasi-linearizable.

On The Sum Of A Sobolev Space And A Weighted $L_P$-Space