The convolution equation P=P*Q of Choquet and Deny and relatively invariant measures on semigroups
Annales de l'Institut Fourier, Tome 21 (1971) no. 4, pp. 87-97.

Choquet et Deny ont considéré la représentation d’une mesure P sur un groupe topologique abélien localement compact comme le produit de convolution de celle-ci par une mesure finie Q:P=P*Q.

Dans cet article, nous essayons de trouver dans le cas de certains semi-groupes localement compacts, les solutions de l’équation précédente, relativement invariantes sur le support de Q. On indique aussi une caractérisation de mesures relativement invariantes sur certains semi-groupes localement compacts. Nos résultats relatifs à l’équation de convolution ont été trouvés par Tortrat pour P finie dans le cas de groupes topologiques quelconques.

Choquet and Deny considered on an abelian locally compact topological group the representation of a measure P as the convolution product of itself and a finite measure Q:P=P*Q.

In this paper, we make an attempt to find, in the case of certain locally compact semigroups, those solutions P of the above equation which are relatively invariant on the support of Q. A characterization of relatively invariant measures on certain locally compact semigroups is also presented. Our results on the above convolution equation, when P is finite, have been obtained also by Tortrat in the case of arbitrary topological groups.

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     author = {Mukherjea, Arunava},
     title = {The convolution equation $P=P^*Q$ of {Choquet} and {Deny} and relatively invariant measures on semigroups},
     journal = {Annales de l'Institut Fourier},
     pages = {87--97},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {21},
     number = {4},
     year = {1971},
     doi = {10.5802/aif.394},
     mrnumber = {48 #9257},
     zbl = {0215.47001},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.394/}
}
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Mukherjea, Arunava. The convolution equation $P=P^*Q$ of Choquet and Deny and relatively invariant measures on semigroups. Annales de l'Institut Fourier, Tome 21 (1971) no. 4, pp. 87-97. doi : 10.5802/aif.394. https://www.numdam.org/articles/10.5802/aif.394/

[A] L. N. Argabright, A note on invariant integrals on locally compact semigroups, Proc. Amer. Math. Soc., 17 (1966), 377-82. | MR | Zbl

[C-D] G. Choquet and J. Deny, Sur l'équation de convolution P = P*Q, C.R. Acad. Sci. Paris, t. 250 (1960), 799-801. | MR | Zbl

[H] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Academic Press, N. Y. (1963).

[M-T] A. Mukherjea and N. A. Tserpes, Idempotent measures on locally compact semigroups Proc. Amer. Math. Soc., Vol. 29, N° 1 (1971), 143-50. | MR | Zbl

[N] K. Numakura, On bicompact semigroups, Math. J. Okayama Univ. 1 (1952), 99-108. | MR | Zbl

[T] A. Tortrat, Lois de probabilité sur un espace topologique complètement régulier et produits infinis à termes indépendants dans un groupe topologique, Ann. Inst. Henri Poincaré, Vol. I, No. 3 (1965), 217-37. | EuDML | Numdam | MR | Zbl

[Ti] A. Tortrat, Lois tendues P sur un demi-groupe topologique complètement simple X, Z. Wahrscheinlichkeitstheorie verw. Geb. 6 (1966), 145-60. | MR | Zbl

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