The author studies natural maps from to for which the image of the normalized Haar measure of is that of . This makes it possible to define probability measures on infinite dimensional unitary groups. A family of integrals relative to these measures are evaluated. Let us state more precisely some of the main results of the paper. For , , or , is the group , or . The author considers the map
for an block representation of . It is proved that the image under the map of the normalized Haar measure of is the Haar measure . Further the author considers the map
where is the upper left block of the matrix , and is the unit ball in the space of matrices over and proves that the image under of is
with and the Lebesgue measure. As an application the integrals
are evaluated in terms of the gamma function.