Abstract: We study shift-invariant nonnegative additive set functions on the real line that differ from the Haar measure by the absence of either the property of countable additivity or the property of \(\sigma\)-finiteness. The Banach measures on the real line defined by Banach limits are finitely additive, nonnegative, and normalized to unity on the entire line. A counting measure taking nonnegative integer values and the value \(+\infty\) is translation-invariant and countably additive, but is not \(\sigma\)-finite. Lebesgue spaces \(L^p\) of functions integrable with respect to translation-invariant measures are introduced. The absence of the strong continuity property of the unitary representation of the shift group in Hilbert spaces of functions square-integrable with respect to the Banach measure is established. A shift-invariant subspace is found, the restriction of the unitary shift group to which is continuous in strong operator theory. It is established that the Fourier transform unitarily maps the subspace of strong continuity onto the space of functions that are square-integrable with respect to the counting measure, but the orthogonal complement of the subspace of continuity lies in the kernel of the Fourier transform.
Funding: Статья подготовлена при финансовой поддержке Министерства науки и высшего образования в~рамках программы Московского центра фундаментальной и прикладной математики по Соглашению № 075-15-2025-345.
For citation: Dzhenzher, E. A. and Sakbaev, V. Zh. Banach Limits, Banach Measure, and Properties of Spaces of Integrable Functions, Vladikavkaz Math. J., 2026, vol. 28, no. 2, pp. 50-69 (in Russian). DOI 10.46698/p4223-2689-0800-t
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