I started reading Katznelson's book on harmonic analysis. on the first page of the book gives the following about the group $\mathbb{T} = \Bbb{R}/2\pi\Bbb{Z}$:
"The Lebesgue measure on $\Bbb{T}$ can be defined by means of the folllowing identification: a function ${f}$ is integrable on $\Bbb{T}$ if the corresponding $2\pi$-periodic function, which we denote again by ${f}$ is integrable on [$0$ , $2\pi$) and we set $$\int_{\Bbb{T}}{f(t)dt}=\int_{0}^{2\pi}{f(x)dx}$$ In other words, we consider [$0$ , $2\pi$) as a model for $\Bbb{T}$ and the Lebesgue measure $dt$ on $\Bbb{T}$ is the restriction of the Lebesgue measure of $\Bbb{R}$ restricted to [$0$ , $2\pi$)."
I'm having trouble understanding how $dt$ is a measure at all. I know I probably just haven't done enough measure theory to understand but i haven't found anything particularly helpful while searching for an answer. If someone could explain this to me or point me in the right direction, it would be greatly appreciated!
The measure is the integral of 1 over the region intersected with (restricted to) $[0,2\pi)$, in other words just the integral of the indicator function of that region over $[0,2\pi)$.