I've come across hyperreal numbers and was curious about something in measure theory. Non measurable sets can be constructed with AC (correct me, if they can also be constructed without AC), and the contradictions these vitali sets induce, are well known. Assuming the measure is non zero, countably many disjoint unions of them would result in infinite measure, and assuming they have 0 measure individually, countable union would imply 0 measure. Now i have a question about defining the measure of some Vitali set $V$ to be an element from the hyperreals, for example $\varepsilon\in \mathbb{R}^*$. I dont know too much about the arithmetic these "numbers" have, but wouldn't it somewhat be intuitive to then have the disjoint union of countably many translates of this set $V$ to be exactly the unit interval, and thus "adding" up to exactly 1 ( length of unit interval). Though, i could see some contradictions arising with this weak definition.
Have similar attempts been made with hyperreal numbers for non measurable sets, and does anyone have some source i could read into? Thanks in advance :)
nonstandard measure theorygives us something called Loeb space, but I don't know anything about nonstandard analysis. – 光復香港 時代革命 Free Hong Kong Apr 07 '21 at 03:19