Well that’s interesting: in order to define unmeasurable sets, you relied on the axiom of choice… I suppose it might be possible to define unmeasurable sets without AC, but maybe not!
Every time I encounter the axiom of choice implying a bunch of crazy stuff, it always loop back to requiring AC. It’s like a bunch of evidence against AC!
I find it interesting that the basic description of AC sounds very plausible, but I’m still convinced mathematicians might have made the wrong choice… (See what i did there? 😄)