Translation invariance of generalized measures on a Hilbert space with continuous test functions

According to Noether’s theorem [1], each continuous symmetry of a physical system corresponds to a certain conservation law. A continuous symmetry is the invariance with respect to a continuous family of transformations. Since invariances and conservation laws in ordinary (finite-dimensional) Euclidean spaces have been systematically studied, we study the translation-invariance of generalized measures from the paper [2] defined in a Hilbert space. Exactly, such invariance is proved in detail for a family of generalized measures introduced in [2], where the properties of these measures are stated without proofs. Invariant generalized measures defined quite differently (with a significantly different space of test functions) have been studied in [3],[4].