There are several way to define a (probability) measure

. The most direct way (the one that is closest to the definition) is by simply giving the measure

of measurable subsets (either of all of them, or a family that would span the

-algebra, like intervals on

). In some situations, it is more natural to give the integral

of measurable functions; letting

gives the usual definition. Both definitions I gave are equivalent. It is however more self-evident in the definition through a measurable function that the measure depends only on the

*set* of eigenvalues, not on an ordering of them, that's why I gave both.