  As a continuation of our previous work on the subject, we prove new measure
invariance results for the Benjamin-Ono equation. The measures are associated
with conservation laws whose leading term is a fractional Sobolev norm of order
larger or equal than 5/2. The new ingredient, compared with the case of
conservation laws whose leading term is an integer Sobolev norm of order larger
or equal than 3 (that has been studied in our previous work), is the use of
suitable orthogonality relations satisfied by multilinear products of centered
complex independent gaussian variables. We also give some partial results for
the measures associated with the two remaining conservation laws at lower
regularity. We plan to complete the proof of their invariance in a separated
article which will be the final in the series. Finally in an appendix, we give
a brief comparing of the recurrence properties of the flows of Benjamin-Ono and
KdV equations.
