Geodesic
On an additive functional homology equation connected with an ergodic rotation of the circle
Izvestiya. Mathematics , Tome 7 (1973) no. 6, pp. 1257-1271.

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It is shown that the title equation with continuous and even analytic right-hand side may have a measurable solution which is not Lebesgue integrable (and even with “worse” properties). 
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D. V. Anosov. On an additive functional homology equation connected with an ergodic rotation of the circle. Izvestiya. Mathematics , Tome 7 (1973) no. 6, pp. 1257-1271. http://geodesic.mathdoc.fr/item/IM2_1973_7_6_a3/

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