On the proximinality of compact operators over spaces of measurable functions

The main results in this paper are, the characterization of all the σ-finite positive measures μ and v, for which K(L1(μ), L1(v)) is proximinal in L(L1(μ), L1(v)); all the σ-finite positive measures μ and v, for which K(L∞(μ), L∞(v)) is proximinal in L(L∞(A), L∞(v)) and all the compact Hausdorff spaces Q, for which K(C(Q), l∞(μ)) is proximinal in L(C(Q), L∞(n)).