First, yes, I know that the Lebesgue outer measure isn't necessarily countably subadditive without the principle of countable choices, and thus isn't an outer measure. Nonetheless, I eagerly anticipate all the answers patiently explaining this to me.
So anyway, to be more precise: given a subset of the real numbers whose Lebesgue submeasure (i.e., a function defined identically to the Lebesgue outer measure, which demonstrably isn't an outer measure in some models of ZF) is infinite, and which fulfills the Carathéodory criterion with respect to the same, is it possible to show in ZF that there exists a subset with arbitrarily large finite Lebesgue submeasure (preferably one that fulfills the Carathéodory criterion, but at all)? I'm pretty sure I can get arbitrarily close to a set of finite submeasure, but I can't seem to extend this to the infinite case. If not, why not?