Measure of the Banach Limit on \(\boldsymbol{L}_{\boldsymbol{\infty}}\boldsymbol{(\mathbb{R})}\)

We study applications of Banach limits in the measure theory. The Banach limit is considered as the composition of the ultrafilter limit and the Cesaro averaging. This construction can be used as the translation-invariant finitely additive measure on an infinite-dimensional Hilbert space. The space of functions that are integrable with respect to the Banach measure is introduced. The Fourier transform in the space of functions that are square integrable with respect to the Banach measure is described. The introduced invariant measure helps to describe both the strong continuity subspace of the Koopman unitary representation of the shift operator and spectral properties of the generator of the shift operator on the invariant subspace of strong continuity.