Skip to main content
Log in

Analogs of the Lebesgue Measure and Diffusion in a Hilbert Space

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript


We study shift-invariant measures on a real separable Hilbert space E, which are also invariant with respect to orthogonal transforms. In this article a finitely additive analogue of the Lebesgue measure is constructed. It is a non-negative finitely additive shift-invariant measure on a special ring of subsets from a space E, which is invariant with respect to orthogonal transforms. The ring contains all infinite-dimensional rectangles, whose products of side lengths are absolutely convergent. We define a Hilbert space \( \mathcal{\mathscr{H}} \), whose elements are complex-valued functions. The functions are square-integrable by some shift-invariant measure, which is also invariant with respect to rotations. We define the expected values of shift operators over random vectors, whose distribution is given by a family of Gaussian measures on a space E. The measures form a semigroup with respect to convolution. We prove that the expected values form a semigroup of contracting self-adjoint operators on a space \( \mathcal{\mathscr{H}} \). The semigroup is not strongly continuous. We also find invariant subspaces, where the semigroup is continuous in the strong operator topology. We investigate the structure of arbitrary (not necessarily continuous) operator semigroups of contracting self-adjoint transforms on a Hilbert space. We show that the suggested in Orlov et al. (Izv. Math. 80(6), 1131–1158 2016) method of the Feynman averaging is applicable to discontinuous semigroups.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Baker, R.: “Lebesgue measure” on \( {R}^{\infty } \). Proc. AMS 113(4), 1023–1029 (1991)

    MATH  Google Scholar 

  2. Borisov, L.A., Orlov, Yu.N., Sakbaev, V.Zh.: Feynman averaging of the semigroups generated by Schrodinger operators. IDAQP 21(2), 1850010 (2018)

  3. Bogachev, V.I.: Gaussian Measures. American Mathematical Society, Rhose Island (1998)

    Book  Google Scholar 

  4. Bogachev, V.I., Krylov, N.V., Röckner, M.: Elliptic and parabolic equations for measures. Russian Math. Surv. 64(6), 973–1078 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  5. Butko, Y.A., Schilling, R.L., Smolyanov, O.G.: Lagrangian and Hamiltonian Feynman formulae for some Feller semigroups and their perturbation. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 15(3), 26–39 (2012)

    Article  MathSciNet  Google Scholar 

  6. Vereschagin, N.K., Shen, A.: Basic Set Theory. [In Russian], MCCME, Moscow (2002)

  7. Vershik, A.M.: Does there exist a lebesgue measure in the Infinite-Dimensional space?. Proc. Steklov Inst. Math. 259, 248–272 (2007)

    Article  MathSciNet  Google Scholar 

  8. Daletskii, Yu.L., Fomin, S.V.: Measures and Differential Equations in Infinite-Dimensional Spaces [In Russian], Nauka, Moscow (1983)

  9. Zavadsky, D.V.: Shift-invariant measures on sequence spaces. Proc. MIPT 9 (4(36)), 142–148 (2017)

    Google Scholar 

  10. Zavadsky, D.V., Viniti, M.: Analogs of Lebesgue measure on the sequences spaces and the classes of integrable functions, Quantum probability, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz. 151, 37–44 (2018)

    Google Scholar 

  11. Orlov, Yu.N., Sakbaev, V.Zh., Smolyanov, O.G.: Unbounded random operators and Feynman formulae. Izv. Math. 80(6), 1131–1158 (2016)

    Article  MathSciNet  Google Scholar 

  12. Sakbaev, V.Zh.: Averaging of random walks and shift-invariant measures on a Hilbert space. Theoret. Math. Phys. 191(3), 886–909 (2017)

    Article  MathSciNet  Google Scholar 

  13. Sakbaev, V.Zh., Viniti, M.: Random walks and measures on Hilbert space that are invariant with respect to shifts and rotations, Differential equations. Mathematical physic. Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz. 140, 88–118 (2017)

    Google Scholar 

  14. Sakbaev, V.Zh., Viniti, M.: Semigroups of operators in the space of function square integrable with respect to traslationary invariant measure on Banach space, Quantum probability. Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz. 151, 73–90 (2018)

    Google Scholar 

  15. Sonis, M.G.: Certain measurable subspaces of the space of all sequences with a Gaussian measure. Uspekhi Mat. Nauk 21(5(131)), 277–279 (1966)

    MathSciNet  Google Scholar 

  16. Remizov, I.D.: Quasi-Feymnan formulas – a method of obtaining the evolution operator for the Schrodinger equation. T. 270(12), 4540–4557 (2016)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to V. Zh. Sakbaev.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The work was supported in the framework of the State Program 5-100

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sakbaev, V.Z., Zavadsky, D.V. Analogs of the Lebesgue Measure and Diffusion in a Hilbert Space. Int J Theor Phys 60, 617–629 (2021).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: