TITLE: Axiom of choice - to use or not to use
QUESTION [31 upvotes]: I was wondering if there are examples of results in mathematics that were first proven using axiom of choice and later someone found a proof of the result without using the axiom of choice.

REPLY [3 votes]: If $G$ is a locally compact abelian group, then there exists a unique measure $\mu$ on the Borel sets of $G$ such that:

$\mu(G)=1$,
For every measurable $A$ and $x\in G$: $\mu(A)=\mu(A+x)$,

We can slightly enlarge the Borel sets to include all the subsets of measure zero sets, and thus have a complete measure.
This measure is called Haar measure. In the case of $\mathbb R$ this is exactly the Lebesgue measure.
Haar (the mathematician) proved its existence on separable compact groups in 1933, and von Neumann proved the uniqueness shortly after. 
The general case for locally compact Abelian groups was proved by Weil and relied on the axiom of choice. Henri Cartan later proved both existence and uniqueness of the general case without the axiom of choice.