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because they don't want to get rid of the positive consequences of assuming the axiom of choice. a huge portion of modern analysis falls apart without it.

in the framework of measure theory, we can simply view the sets produced in the banach tarski paradox as nonmeasurable. in fact, it is a result of solovay that nonmeasurable sets do not exist in ZF without the assumption of AC.

another interesting consequence of AC: it is trivial using the axiom of choice to construct a discontinuous additive function using a hamel basis for R over Q. we can also prove that measurable additive functions are linear (taking the convolution of exp(if(x)) with the standard mollifier when f is measurable + additive). again without AC, all discontinuous additive functions no longer exist, which is interesting because their existence doesn't seem to be related to measure theory in the first place.