Most people who learn measure theory secretly believe that all sets are “really” measurable. The one example of a nonmeasurable set anyone sees is that of Vitali, which requires the axiom of choice, so it’s tempting to believe that without the axiom of choice, every set is measurable. This belief is only reenforced by a result of Solovay’s that the axioms of set theory, removing the axiom of choice, and adding the axiom that every set is measurable remain consistent.
Pace the Solovay result, this belief is not quite right. Within set theory it is possible to construct sets without using the axiom of choice which are not necessarily measurable. The sets are not exactly nonmeasurable, but their measurability is independent of ZFC. (Solovay proved that their measurability can be added as a new axiom.)
These sets can be constructed as follows. The image of a Borel set under a continuous function is known as an
analytic set. Analytic sets are measurable. The complement of an analytic set (a coanalytic set) is also measurable, but the image of a coanalytic set under a continuous function in general has undecidable measurability.
This raises the metamethematical question, should they be measurable? My (vague) intuition is no: whether or not we assert the measurability of these sets, we have no way to actually assign a measure to them. Woodin, in his survey article about the continuum hypothesis that we linked to earlier argues yes.