Applying an invariant measure on phase space, we study the Koopman representation of a group of symplectomorphisms in an infinite-dimensional Hilbert space equipped with a translation-invariant symplectic form. The phase space is equipped with a finitely additive measure, invariant under the group of symplectomorphisms generated by Liouville-integrable Hamiltonian systems. We construct an invariant measure of Lebesgue type by applying a special countable product of Lebesgue measures on real lines. An invariant measure of Banach type is constructed by applying a countable product of Banach measures (defined by the Banach limit) on real lines. One of the advantages of an invariant measure of Banach type compared to an invariant measure of Lebesgue type is finiteness of the values of this measure in the entire space. The introduced invariant measures help us to describe both the strong continuity subspaces of the Koopman unitary representation of an infinite-dimensional Hamiltonian flow and the spectral properties of the constraint generator of the unitary representation on the invariant strong continuity subspace.

应用相空间上的不变测度，我们研究了具有平移不变辛形式的无限维希尔伯特空间中一组辛同胚的库普曼表示。相空间配备了有限加性测度，在由刘维尔可积哈密顿系统生成的辛同态群下保持不变。我们通过在实线上应用勒贝格测度的特殊可数乘积来构造勒贝格类型的不变测度。 Banach 类型的不变测度是通过在实线上应用 Banach 测度（由 Banach 极限定义）的可数乘积来构造的。与 Lebesgue 类型的不变测度相比，Banach 类型的不变测度的优点之一是该测度在整个空间中的值的有限性。引入的不变测度帮助我们描述无限维哈密顿流的库普曼酉表示的强连续子空间以及不变强连续子空间上酉表示的约束生成器的谱特性。