Induced ideals in Cohen and random extensions

We show that the following is consistent (relative to the consistency of a measurable cardinal): There is no real valued measurable cardinal below continuum and there is a finitely additive extension m:P([0,1])→[0,1] of Lebesgue measure whose null ideal is a sigma ideal. We also show that there is a countable partition of [0,1] into interior free sets under the m-density topology of any such extension.