  Modulo some natural generalizations to noncompact spaces, we show in this
letter that Moyal planes are nonunital spectral triples in the sense of Connes.
The action functional of these triples is computed, and we obtain the expected
result, ie the noncommutative Yang-Mills action associated with the Moyal
product. In particular, we show that Moyal gauge theory naturally fit into the
rigorous framework of noncommutative geometry.
