Category Archives: set-theory

A set that is not Lebesgue measurable

The Lebesgue measure $\lambda$ on the real line is a countably additive measure that assigns to each interval $[a,b]$ with $a \leq b$ its length $b-a$. Why construct the Lebesgue measure? It’s so that we can get around a blemish of the Riemann integral: namely, that the Riemann integration theory does not know how to […]

Every Set Has a Group Structure Iff Axiom of Choice

Here I explain the proof that in ZF, the axiom of choice (AC) is equivalent to every nonempty set having group structure (GS). I first learned of the nontrivial direction of this argument in this MathOverflow post and as far as I know first appeared in “Some new algebraic equivalents of the axiom of choice” […]