Ultraproducts of real interpolation spaces between Lp-spaces

Let {(L-p0 (Omega(d), mu(d)), L-p1 (Omega(d), mu(d))), d is an element of D}, 1 <= p(0) < p(1) < infinity, be a family of compatible couples of L-p-spaces. We show that, given a countably incomplete ultrafilter U in D, the ultraproduct ((L-p0 (Omega(d), mu(d)), L-p1 (Omega(d), mu(d)))(theta),(q))(U), 0 < theta < 1, 1 <= q < infinity of interpolation spaces defined by the real method is \isomorphic to the direct sum of an interpolation space of type (L-p0 (Omega(1), v(1)), L-p1 (Omega(1), v(1)))(theta),q, an intermediate Kothe space between e(p0)(Omega(2), v(2)) and e(p1)(Omega(2), v(2)), (Omega(2), v(2)) being a purely atomic measure space, and a Kothe function space K (Omega(3)) defined on some purely non atomic measure space (Omega(3), v(3)) in such a way that Omega(2) boolean OR Omega(3) not equal 0.