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I am looking for references, measure-integration theory where the $d$-dimensional torus $\mathbb{T}^d$ is treared rigorously: borel $\sigma$-algebra, measure functions, measures on $(\mathbb{T}^d,\mathcal{B}(\mathbb{T}^d)),$ I looked in Folland real analysis book, but nothing from the above is explained, Also Rudin real and complex analysis doesn't incluse the above.

Are you aware of any reference treating these subjects?

mathex
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    See surface measure and Gauss-Green theorem proof. There, I introduce the Riemann-Lebesgue measure on any semi-Riemannian manifold. In particular, for every Riemannian manifold. The (round) circle is naturally a Riemannian manifold, so it inherits a measure; the torus being the product inherits a measure as well. Or you can view the torus as a Riemannian manifold in its own right and get a measure that way. There’s also a general theory of measures on Lie groups (topological groups) which you may want to google. – peek-a-boo May 20 '24 at 22:28
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    if you search through the (sub)links, you’ll find that I have several answers about integration on manifolds: (curved)Fubini, change-of-variables, Radon-Nikodym derivatives, connecting with the Riesz-approach to integration and measure etc. Oh and here is an answer about the Torus specifically. – peek-a-boo May 20 '24 at 22:31

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