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Let G be a Lie supergroup and H a closed subsupergroup. We study theunimodularity of the homogeneous supermanifold G/H, i.e. the existence ofG-invariant sections of its Berezinian line bundle. To that end, we expressthis line bundle as a G-equivariant associated bundle of the principal H-bundleG over G/H. We also study the fibre integration of Berezinians on orientedfibre bundles. As an application, we prove a formula of `Fubini' type: theinvariant integral over G can be expressed (up to sign) by a succesiveinvariant integration over H and G/H. Moreover, we derive analogues of integralformulae for the transformation under local isomorphisms of unimodularhomogeneous superspaces G/H and S/T, and under the products of subsupergroupsof Lie supergroups. The classical counterparts of these formulae have numerousapplications in harmonic analysis.

Invariant Berezin integration on homogeneous supermanifolds