Who first proved ergodicity of irrational rotations of the circle?

The idea to study what we call irrational rotation of a torus indeed belongs to Nicole Oresme, at least he clearly understood the density of the trajectories (which is not the same as ergodicity or equidistribution! One thing is to say that a trajectory visits every interval infinitely many times, and another thing to say HOW OFTEN. Oresme has absolutely no adequate language or tools to address, or even to state the second question). He uses the words such as "probability" too, for example he tries to say that a random number is almost surely irrational. But he really has no clear concept of a real number. With a hindsight one can read in his paper more than he really wrote. The first rigorous proof, according to the modern standards belongs to Hermann Weyl, and on my opinion it is an enormous overstatement to credit ergodicity to Oresme.

References: These two papers contain the condensed English translation with extended comments of Oresme two texts on irrational rotation:

Edward Grant, Oresme and His De Proportionibus Proportionum, Isis, Vol. 51, No. 3 (Sep., 1960), pp. 293-314, doi: 10.1086/348912, jstor.

Edward Grant, Nicole Oresme and the Commensurability or Incommensurability of the Celestial Motions, Archive for History of Exact Sciences Vol. 1, No. 4 (26.10.1961), pp. 420-458, doi: 10.1007/BF00328576, jstor.

Unlike the papers of Oresme, Weyl's paper is only available in German and Russian: Ueber die Gleichverteilung von Zahlen mod. Eins,". Math. Ann. 77 (3): 313–352, doi: 10.1007/BF01475864, eudml.

Вейль Г, Избранные труды. Математика. Теоретическая физика, М.: Наука, 1984

By the way, Oresme's main goal was to refute astrology by showing that celestial phenomena, like conjunctions and oppositions are essentially random:-) Because the periods are incommensurable. Because two random numbers are incommensurable. Exciting reading! Written around 1360. And one of the main goals of Weyl was to prove the existence of Mean Motion (conjectured by Lagrange), another problem coming from astronomy. Weyl's proof of Lagrange's conjecture contained a gap which was filled by Tornehave and Jessen in 1943. (I added this because these remarkable results seems to be almost forgotten).

the proof goes back to Nicole Oresme in his paper De commensurabilitate vel incommensurabilitate motuum celi [On the Commensurability or Incommensurability of the Motions of the Heavens], dated around 1360, see

• Nicole Oresme and the commensurability or incommensurability of celestial motions (contains an annotated English translation of Oresme's Latin text)
• Oresme's Proof of the Density of Rotations of a Circle through an Irrational Angle
• Nicole Oresme and the ergodicity of rotations

Weyl had only one "big" paper on uniform distribution theory, which is the 1916 paper. However, the fact that $(n \alpha)$ is equidistributed for irrational $\alpha$ is usually attributed to Bohl, Spierpinski and Weyl (independently), who proved it in 1909-1910. However, they did not prove anything about ergodicity. The ergodicity question does not ad-hoc have any close relation to the equidistribution problem.