Measurability of Curved Bars on a Hilbert Space

We introduce a class of sets measurable with respect to a translation-invariant measure on a Hilbert space that are representable as a set of infinite-dimensional fibers over a finite-dimensional base. It is established that if the base (i.e., the orthogonal projection of a set onto some finite-dimensional subspace) is Jordan measurable and each fiber over a point of the base is an infinite-dimensional bar with Lipschitz continuity with respect to the point, then such a set, called a $$Lip(n)$$ -curved bar, is measurable with respect to a translation-invariant measure. We define a group of transformations of a Hilbert space under which the introduced class of $$Lip(n)$$ -curved bars is invariant.