[dahart]

Right from that article you linked: "A non-self-intersecting continuous curve cannot fill the unit square because that will make the curve a homeomorphism from the unit interval onto the unit square (any continuous bijection from a compact space onto a Hausdorff space is a homeomorphism). But a unit square has no cut-point, and so cannot be homeomorphic to the unit interval, in which all points except the endpoints are cut-points."

[waqf]

Yeah, a non-self-intersecting map cannot, but OP didn't specify that, only "epimorphic" which certainly can.

Moreover OP's argument specifically proves too much, because space-filling curves (as described in the article) have a range with positive Borel measure.