TITLE: Measurable cardinals under Axiom of Determinacy
QUESTION [9 upvotes]: I seem to remember reading somewhere that ZF+AD proves that $\omega_1$ and $\omega_2$ are measurable cardinals.
Is that right?
If so, can someone [point me to or give here] a [sketch or proof] of these results?

REPLY [2 votes]: For a different proof of the measurability of $\omega_1$ under AD, you could look at Eugene Kleinberg's book Infinitary Combinatorics and the Axiom of Determinateness. He shows it using the strong partition relation  on $\omega_1$(that any function colouring the $\omega_1$-cardinality subsets of $\omega_1$ into 2 colours has a homogeneous subset of size $\omega_1$), showing that the filter of $\omega$-closed unbounded sets of $\omega_1$ is actually a normal ultrafilter. The only issue is that proving the strong partition relation is slightly involved. 
P.S. Could someone please tell me how to get the arrow notation for representing partition relations right? I have absolutely no idea. Also, please correct me if I'm wrong above; it's my first post on Math Overflow!